THE  LIBRARY 
OF 

THE  UNIVERSITY 

OF  CALIFORNIA 

LOS  ANGELES 


THE 

TEMPERATURE-ENTROPY 
DIAGRAM 


BY 

CHARLES  W.   BERRY 

Assistant  Professor  of  Mechanical  Engineering  in  the 
Massachusetts  Institute  of  Technology 


THIRD  EDITION,  REVISED  AND  ENLARGED. 
FIRST  THOUSAND 


NEW  YORK 

JOHN  WILEY   &   SONS 

LONPON  :  CHAPMAN  &  HALL,  LIMITED 

1911 


Copyright,  1905,  1908,  1911, 

BY 

CHARLES  W.  BERRY 


THE  SCIENTIFIC   PRESS 
tRT  OBUMMONO 
BROOKLYN, 


TT 


II 

PREFACE   TO   THIRD  EDITION 


THE  present  revision  includes  the  following  additions 
and  changes:  Minor  insertions  have  been  made  in  the 
chapters  upon  the  Flow  of  Fluids,  the  Gas  Engine  Cycles, 
and  the  Non-conducting  Steam  Engine.  The  chapter 
on  Refrigeration  and  the  Warming  Engine  has  been  ex- 
panded into  separate  chapters  upon  each  subject.  A 
special  chapter  has  been  added  upon  Entropy  Analysis 
in  the  Boiler  Room.  The  Tables  upon  the  Efficiency, 
'Water  and  Heat  Consumption  of  the  Rankine  Cycle 
have  been  extended  to  cover  the  range  of  low-pressure 
turbines  as  well  as  high-pressure  reciprocating  engines. 
All  illustrative  problems  have  been  recalculated  to 
agree  with  the  most  recent  and  accurate  data  upon 
steam.  The  second  and  third  editions  of  this  book 
have  so  extended  its  scope  that  it  is  now  a  treatise 
upon  graphical  thermodynamics  although  still  abiding 
by  the  limitations  imposed  by  its  title. 

CHARLES  W.  BERRY. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
January,  1911. 

iii 


PREFACE   TO    SECOND   EDITION. 


IN  the  revised  edition  of  the  Temperature-Entropy 
Diagram  a  more  extended  application  of  the  principles 
of  the  T^-analysis  to  advanced  problems  of  thermo- 
dynamics has  been  made  than  was  possible  in  the 
limited  scope  of  the  previous  edition.  The  Chapter 
on  the  Flow  of  Fluids  has  been  entirely  rewritten  and 
treats  at  length  various  irreversible  processes.  A 
graphical  method  of  projecting  from  the  pv-  into 
the  T^-plane  has  been  elaborated  for  perfect  gases 
and  its  application  illustrated  in  the  chapters  on 
Hot-air  Engines  and  Gas-engines.  The  various  factors 
affecting  the  cylinder  efficiency  of  both  gas-  and 
steam-engines  have  been  thoroughly  discussed.  One 
chapter  has  been  devoted  to  the  thermodynamics  of 
mixtures  of  gases  and  vapors,  and  another  to  the 
description  and  use  of  Mollier's  total  energy-entropy 
diagram. 

CHARLES  W.  BERRY. 
MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 

October,  1908.  iv 


PREFACE   TO   THE   FIRST   EDITION. 


Tms  little  volume  was  prepared  for  the  use  of  stu- 
dents of  thermodynamics,  and  therefore  I  have  en- 
deavored to  bring  together  hi  logical  order  certain 
information  concerning  the  construction,  interpreta- 
tion, and  applications  to  engineering  problems  of  the 
temperature-entropy  diagram,  which  otherwise  would 
not  be  readily  available  for  them,  as  such  information 
is  scattered  throughout  various  treatises.  The  book  is 
not  intended  for  the  advanced  student,  as  he  is  already 
familiar  with  its  contents,  neither  is  it  expected  that 
one  entirely  ignorant  of  thermodynamics  can  use  it  to 
advantage,  as  the  reader  is  assumed  to  have  an  ele- 
mentary knowledge  of  the  fundamental  theory  and 
equations.  An  exhaustive  treatment  has  not  been 
attempted,  but  it  is  believed  that  the  graphical  presen- 
tation here  given  will  aid  the  student  to  a  clearer 
comprehension  of  the  fundamental  principles  of  thermo- 
dynamics and  make  it  possible  for  him  to  read  under- 
standingly  more  pretentious  works. 

CHARLES  W.  BERRY. 

MASSACHUSETTS  INSTITUTE  OF  TECHNOLOGY, 
January,  1905. 


CONTENTS. 


PACHE 

INTRODUCTION ix 

TABLE  OF  SYMBOLS  USED  IN  TEXT xvii 

CHAPTER 

I.  GENERAL   DISCUSSION.     REVERSIBLE    PROCESSES   AND 

CYCLES.     EFFECT  OF  IRREVERSIBILITY  -  , 1 

II.  THE  TEMPERATURE-ENTROPY   DIAGRAM   FOR   PERFECT 

GASES 13 

III.  THE  TEMPERATURE-ENTROPY  DIAGRAM  FOR  SATURATED 

STEAM 43 

IV,  THE  TEMPERATURE-ENTROPY  DIAGRAM  FOR  SUPER- 

HEATED VAPORS 65 

V.  THE  TEMPERATURE-ENTROPY  DIAGRAM  FOR  THE  FLOW 

OF  FLUIDS  > 78 

VI.  MOLLIER'S  TOTAL  ENERGY-ENTROPY  DIAGRAM 131 

VII.  THERMODYNAMICS  OF  MIXTURES  OF  GASES,  OF  GASES 

AND  VAPORS,  AND  OP  VAPORS 139 

VIII.  THE    TEMPERATURE- ENTROPY    DIAGRAM    APPLIED    TO 

HOT-AIR  ENGINES 158 

IX.  THE  TEMPERATURE-ENTROPY  DIAGRAM  APPLIED  TO  GAS- 
ENGINE  CYCLES 171 

X.  THE  GAS-ENGINE  INDICATOR  CARD 206 

XI.  THE  TEMPERATURE-ENTROPY  DIAGRAM  APPLIED  TO  THE 

NON-CONDUCTING  STEAM-ENGINE 224 

XII.  THE  MULTIPLE-FLUID  OR  WASTE-HEAT  ENGINE 248 

XIII.  THE  TEMPERATURE-ENTROPY  DIAGRAM  OF  THE  ACTUAL 

STEAM-ENGINE  CYCLE 258 

vii 


viii  CONTENTS. 


CHAl 


PAGE 


XIV.  STEAM-ENGINE  CYLINDER  EFFICIENCY 276 

XV.  LIQUEFACTION  OF  VAPORS  AND  GASES 294 

XVI.  APPLICATION  OF  THE  TEMPERATURE-ENTROPY  DIAGRAM 

TO  AIR-COMPRESSORS  AND  AIR-MOTORS 302 

XVII.  DISCUSSION  OF  REFRIGERATING  PROCESSES 337 

XVIII.  DISCUSSION  OF  KELVIN'S  WARMING  ENGINE 359 

XIX.  ENTROPY  ANALYSIS  IN  THE  BOILER-ROOM 374 

TABLE  OF  PROPERTIES   OF  SATURATED  STEAM   FROM   400°  F. 

TO  THE   CRITICAL  TEMPERATURE 388 

HYPEBBOLIC  OB  NAPERIAN  LOGARITHMS  ...  389 


INTRODUCTION. 


IT  seems  necessary  in  a  book  dealing  with  the  appli- 
cation of  the  temperature-entropy  diagram  to  discuss 
briefly  that  "  ghostly  quantity,"  entropy,  although  I 
do  not  intend  to  give  any  new  definition  of  a  function 
already  too  variously  defined,  but  rather  to  pick  out 
such  of  the  present  ones  as  are  correct. 

One  has  but  to  plot  an  irreversible  adiabatic  process 
hi  the  temperature-entropy  plane  to  realize  once  and 
for  all  that  the  entropy  does  not  necessarily  remain 
constant  along  an  adiabatic  line.  In  fact  isentropic 
and  adiabatic  changes  coincide  only  when  the  latter 
process  is  reversible:  and  such  a  change  practically 
never  occurs  in  nature.  For  example,  in  one  irrevers- 
ible adiabatic  expansion  representing  the  flow  through 
a  non-conducting  porous  plug,  the  heat  added  is  zero, 

/7/-) 
-~-  =  0,    but    nevertheless    the   entropy   of 

the  substance  increases.  It  is  even  possible  to  imagine 
an  irreversible  process  which  is  at  the  same  time  isen- 
tropic. Suppose  a  gas  to  expand  through  a  nozzle 

iz 


X  INTRODUCTION. 

losing "  heat  by  radiation  and  conduction  and  also 
undergoing  friction  losses  whereby  part  of  its  kinetic 
energy  is  dissipated  and  restored  to  the  gas  as  heat. 
The  loss  of  heat  by  radiation  and  conduction  will  re- 
duce the  entropy  of  the  gas,  while  the  gain  of  heat  by 
friction  will  increase  it.  It  is  possible  to  consider 
these  two  opposing  influences  as  equal,  and  then  the 
flow  will  be  isen tropic  although  not  adiabatic. 

The  entropy  of  a  substance,  just  as  much  as  its 
mtrinsic  energy,  specific  volume,  specific  pressure,  or 
temperature,  has  a  definite  value  for  each  position  of 
the  state  point  upon  the  characteristic  surface,  and 
the  increase  in  the  value  of  the  entropy  in  changing 
from  one  point  to  another  is  a  definite  quantity  regard- 
less of  the  path  chosen.  The  magnitude  of  this  increase 

is  equal  to  /  -~-,  taken  along  any  reversible  path 
between  these  points.  This  fact  has  led  to  the  inexact 
definition  of  change  of  entropy,  d<})  =  -~-,  a  definition 

true  only  for  ideal  reversible  processes  and  hence 
utterly  wrong  when  applied  to  actual  irreversible  pro- 
cesses, as  in  general  d</>>-~-. 

Since  for  reversible  cycles  d<£  =  -~-,  it    follows   that 

the  heat  added  during  any  reversible  change  is  equal 

f2 
to     Q  =   /    Td<p,    and     for     an     isothermal     process 


INTRODUCTION.  xi 

(}>i}.  This  is  undoubtedly  the  basis  for  all 
the  physical  analogies  attempting  to  explain  entropy 
as  heat-weight,  etc.,  and  also  for  the  name  "heat 
diagram"  applied  to  the  temperature-entropy  diagram. 
The  area  under  a  curve  in  the  7^-plane  is  equal  to 
the  heat  received  from,  or  rejected  to,  some  outside 
body  only  when  the  process  is  reversible. 

Similarly  if  the  specific  pressure  and  specific  volume 
of  a  gas  could  be  ascertained  at  various  points  in  its 
passage  through  a  porous  plug,  these  points  if  plotted 
would  form  a  pr-curve  giving  a  true  history  of  the 
movement  of  the  state  point,  but  the  area  under  the 
curve  would  not  represent  work,  as  no  external  work 
has  been  performed. 

Preston  in  his  Theory  of  Heat  says:  "The  entropy 
of  a  body  being  taken  arbitrarily  as  zero  in  some  stand- 
ard condition  A,  defined  by  some  standard  temperature 
and  pressure  (or  volume),  the  entropy  in  any  other 

/A(~\ 
-f=f  taken  along  any  reversible 

path  by  which  the  body  may  be  brought  to  B  from  the 
standard  state  A.  The  path  may  obviously  be  an  arc 
AG  of  an  isothermal  line  passing  through  the  point 
defining  the  standard  state,  together  with  the  arc  BC  of 
the  adiabatic  line  passing  through  B.  The  entropy  in 
the  state  B  may  consequently  be  measured  thus.  Let 
the  volume  be  changed  adiabatically "  (reversible 
process)  "until  the  standard  temperature  T  is  attained, 


INTRODUCTION. 


and  then  change  the  volume  isothermally  until   the 
standard  pressure  is  attained.     If  the  quantity  of  heat 


imparted  during  the  latter  operation  be  Q,  the  entropy 

Q 
in  the  state  B  is  <f>  =  ip 

"  In  this  operation  the  temperature  and  pressure  are 
supposed  uniform  throughout  the  body.  ...  If,  how- 
ever, any  body  be  subject  to  operations  which  produce 
inequalities  of  temperature  in  the  mass,  there  will 
be  a  transference  of  heat  from  the  warmer  to  the  colder 
parts  by  conduction  and  radiation,  and  although  the 
body  may  neither  receive  heat  from  nor  give  it  out  to 
other  bodies  (so  that  the  transformation  is  adiabatic 
throughout),  yet  on  account  of  the  inequalities  of  tem- 
perature, the  entropy  of  the  mass  will  increase,  .  .  . 
and  under  these  circumstances  the  transformation  will 
not  be  isentropic." 

Swinburne  in  his  Entropy  says:  "Entropy  may  be 
defined  thus:  Increase  of  entropy  is  a  quantity  which, 
when  multiplied  by  the  lowest  available  temperature, 


INT  ROD  UC  TION.  xiii 

gives  the  incurred  waste.  In  other  words,  the  increase  of 
entropy  multiplied  by  the  lowest  temperature  available 
gives  the  energy  that  either  has  been  already  irrevocably 
degraded  into  heat  during  the  change  in  question,  or 
must,  at  least,  be  degraded  into  heat  in  bringing  the 
working  substance  back  to  the  standard  state.  .  .  . 

"Thus  the  entropy  of  the  body  in  state  B  is  not  a 
function  of  the  heat  actually  taken  in  during  its  change 
from  A  to  B,  as  the  change  must  have  been  partially, 
and  may  have  been  wholly,  irreversible;  but  it  can  be 
measured  as  a  function  of  the  heat  which  would  have 
to  be  taken  in  to  change  from  A  to  B  reversibly,  or 
which  would  have  to  be  given  out  if  the  substance  were 
changed  from  B  to  A  reversibly,  which  amounts  to 
the  same  thing.  .  .  . 

"The  entropy  of  a  body  therefore  depends  only  on  the 
state,  and  not  on  its  past  history." 

Planck  in  his  Treatise  on  Thermodynamics  writes 
(see  English  translation  by  Ogg): 

"A  process  which  can  in  no  way  be  completely 
reversed  is  termed  irreversible,  all  other  processes  re- 
versible. That  a  process  may  be  irreversible,  it  is  not 
sufficient  that  it  cannot  be  directly  reversed.  This 
is  the  case  with  many  mechanical  processes  which  are 
not  irreversible.  The  full  requirement  is,  that  it  be 
impossible,  even  with  the  assistance  of  all  agents  in 
nature,  to  restore  everywhere  the  exact  initial  state 
when  the  process  has  once  taken  place.  .  .  .  The  gen- 


xiv  INTRODUCTION. 

oration  of  heat  by  friction,  the  expansion  of  a  gas 
without  the  performance  of  external  work,  and  the 
absorption  of  external  heat,  the  conduction  of  heat, 
etc.,  are  irreversible  processes. 

"Since  there  exists  in  nature  no  process  entirely 
free  from  friction  or  heat-conduction,  all  processes 
which  actually  take  place  in  nature,  if  the  second 
law  be  correct,  are  in  reality  irreversible;  reversible 
processes  form  only  an  ideal  limiting  case.  They  are, 
however,  of  considerable  importance  for  theoretical 
demonstration  and  for  application  to  states  of  equilib- 
rium. 

"//  a  homogeneous  body  be  taken  through  a  series 
of  states  of  equilibrium,  that  follow  continuously  from 
one  another,  back  to  its  initial  state,  then  the  summation 

of  the  differential fj*~  extending  over  all  the  states 

of  that  process  gives  the  value  zero.  It  follows  that,  if 
the  process  be  not  continued  until  the  initial  state,  1, 
is  again  reached,  but  be  stopped  at  a  certain  state,  2, 

the   value   of   the   summation    /      — ~—      depends 

only  on  the  states  1  and  2,  not  on  the  manner  of  the 
transformation  from  state  1  to  state  2.  ... 

"The  (above)  integral  .  .  .  has  been  called  by 
Clausius  the  entropy  of  the  body  in  state  2,  referred 
to- state  1  as  the  zero  state.  The  entropy  of  a  body  in 
a  given  state,  like  the  internal  energy,  is  completely 


INTRODUCTION.  XV 

determined  up  to  an  additive  constant,  whose  value 
depends  on  the  zero  state. 

"It  is  impossible  in  any  way  to  diminish  the  entropy 
of  a  system  of  bodies  without  thereby  leaving  behind 
changes  in  other  bodies.  If,  therefore,  a  system  of 
bodies  has  changed  its  state  in  a  physical  or  chemical 
way,  without  leaving  any  change  in  bodies  not  belong- 
ing to  the  system,  then  the  entropy  in  the  final  state 
is  greater  than,  or,  in  the  limit,  equal  to  the  entropy 
in  the  initial  state.  The  limiting  case  corresponds  to 
reversible,  all  others  to  irreversible,  processes. 

' '  The  restriction  .  .  .  that  no  changes  must  remain 
in  bodies  outside  the  system  is  easily  dispensed  with  by 
including  in  the  system  all  bodies  that  may  be  affected 
in  any  way  by  the  process  considered.  The  proposition 
then  becomes: 

"  Every  physical  or  chemical  process  in  nature  takes 
place  in  such  a  way  as  to  increase  the  sum  of  the  entropies 
of  all  the  bodies  taking  any  part  in  the  process.  In  the 
limit,  i.e.  for  reversible  processes,  the  sum  of  the  entropies 
remains  unchanged.  This  is  the  most  general  state- 
ment of  the  second  law  of  Thermodynamics." 


SYMBOLS  USED  IN  THE  FOLLOWING  PAGES, 


A  =y  =Heat  equivalent  of  a  unit  of  work. 

Apw=Heat  equivalent  of  the  external  work  of  vaporization. 

c  =  General  expression  for  the  specific  heat  during  any  change, 
cp=Specific  heat  at  constant  pressure. 
cv  =  Specific  heat  at  constant  volume. 

J=  Change  of  ... 
#=  Internal  (intrinsic)  energy  in  work  units  =S+I. 

T)  =  Thermal  efficiency  of  an  engine. 
F=Area  (Flache). 
G=  Weight  (Gewichf). 

g=  Acceleration  due  to  gravity. 
H=  Total   heat  above  some  arbitrary  zero,    =q,  q+xr,    A, 

A  +  Cpfc-0. 

h  =  Specific  heat  of  dry  saturated  vapor. 

/  =  Internal  energy  due  to  separation  of  molecules. 

t  =•  Total  energy  =  A  (E+pv)  =  II  +  Apn. 

k  =  —  for  a  perfect  gas. 

A  =  Total  heat  of  dry  saturated  vapor  =q+r. 

n  =  Exponent  of  the  poly  tropic  change  pvn  =c. 

p=Specific  pressure. 

<£=  General  expression  for  entropy. 

Q=Heat  received  from  or  exhausted  to  some  outside  body. 

5=  Heat  of  the  liquid. 

R=7p,  for  a  perfect  gas. 
r  =Total  latent  heat  of  vaporization  =  o  +  Apu. 


SYMBOLS   USED  IN  THE  FOLLOWING  PAGES. 

/o  =  Internal  latent  heat  of  vaporization. 
m  =  Entropy  of  vaporization. 

5  =  Internal  energy  due  to  vibration  of  molecules, 
s  =  Specific  volume  of  a  dry  saturated  vapor. 

<7=  Specific  volume  of  a  liquid. 

77=  Temperature  in  degrees  absolute,  Fahrenheit  or  Centi- 
grade. 

t  =  Temperature  in  degrees  Fahrenheit  or  Centigrade. 
tg  =  Temperature  of  superheated  vapor. 

6  =  Entropy  of  the  liquid  =  J  -^r. 

u=  Increase  of  volume  due  to  vaporization  =s  —  a. 
v  =  General  value  of  specific  volume  =  a.  xu  +  a,  s,  etc. 
V  =  Velocity  in  linear  units  per  second. 

y2 

2~  =  Kinetic  energy  of  a  jet  in  work  units. 

W  =Work  performed  by  or  upon  a  substance. 
x  =  Quality  of  a  unit  weight  of  a  mixture  of  a  liquid  and  its 
vapor. 


THE    TEMPERATURE-ENTROPY 
DIAGRAM. 


CHAPTER  I. 

GENERAL  DISCUSSION.     REVERSIBLE  PROCESSES  AND 
CYCLES.     EFFECT  OF  IRREVERSIBILITY. 

THE  condition  of  a  substance  is  in  general  completely 
denned  by  any  two  of  its  five  characteristic  properties, 
specific  pressure,  specific  volume,  absolute  temperature, 
intrinsic  energy,  and  entropy.  The  relations  existing 
between  any  three  of  these  qualities  being  expressed 
by  the  formula  z  =  f(x,  y}.  Exceptions  to  this  occur 
in  the  case  of  saturated  vapors,  where  specific  pressure 
and  temperature  alone  do  not  suffice  to  define  com- 
pletely the  state  of  the  body,  and  in  the  case  of  per- 
fect gases,  where  isodynamic  and  isothermal  changes 
•coincide. 

In  the  analytical  solution  of  thermodynamic  problems 
those  formula?  are  used  which  contain  the  properties 
most  important  for  the  investigation  in  hand. 
Similarly  in  graphical  solutions  any  pair  of  character- 


2  THE   TEMPERATURE-ENTROPY  DIAGRAM. 

istics  may  be  used  as  coordinates  as  convenient,  and 
then  the  curves,  y  =  j(x),  expressing  the  relations 
between  the  various  characteristics  drawn  in  this 
r?/-plane,  assume  different  forms  according  to  the 
laws  of  variation  of  the  five  variables  p,  v,  T,  E,  and  <£. 

The  pressure-volume  diagram,  or  7W-diagram,  is  the 
one  most  widely  used,  as  its  coordinates  are  those 
common  to  every-day  experience,  but  for  some  investi- 
gations of  heat-transference,  changes  of  temperature, 
changes  of  '  entropy,  etc.,  the  temperature-entropy 
diagram,  or  the  T  ^-diagram,  lends  more  ready  assist- 
ance. 

As  a  consequence  of  the  fundamental  relation 
2==/(x,  y),  any  curve  p^f^v)  in  the  py-plane  has  its 
counterpart  ^>  =  /2(7T)  in  the  T<£-plane;  both  being  but 
special  projections  of  the  same  change  on  the  charac- 
teristic surface. 

In  the  following  a  discussion  will  be  given  of  the 
different  forms  assumed  by  the  curve  <p=f(T),  in  the 
case  of  perfect  gases,  saturated  steam,  and  superheated 
steam  under  the  conditions  of  constant  temperature, 
constant  entropy,  constant  pressure,  constant  volume, 
constant  intrinsic  energy,  etc.,  and  also  a  considera- 
tion of  the  interpretation  to  be  given  to  the  areas 
included  between  any  given  curve  and  the  axis,  together 
with  certain  other  interesting  details. 

If  0(j)  and  OT  (Fig.  1)  represent  the  axis  of  entropy 
and  temperature  respectively,  any  isothermal  change, 


REVERSIBLE   PROCESSES. 


as  at  the  temperature  T3,  will  be  represented  by  a  hori- 
zontal line  ab,  and  similarly  any  isentropic  change 
(or  adiabatic  in  the  case  of  a  reversible  process)  will 
be  represented  by  a  vertical  line,  as  cd.  The  forms  of 
the  curves  for  constant  pressure,  etc.,  will  vary  with 


FIG.  1. 

the  state  of  the  substance  and  will  be   investigated 
for  each  special  case. 

Let  AB  represent  any  reversible'  process  and  let  c 
be  the  specific  heat  of  the  substance  during  this  change. 
In  order  to  increase  the  temperature  of  a  unit  weight 
of  the  substance  by  the  amount  dT  there  will  be  neces- 
sary the  expenditure  of  the  heat  dQ=cdT.  But  for 
a  reversible  cycle  there  exists  the  further  relation 

d<{>  =-,  or  dQ  =  Td$,  and  therefore 


or 


Q 


rr* 
I     cdT 

jTi 


Td!>. 


THE   TEMPERATURE-ENTROPY  DIAGRAM. 

Now  I      Tdd>    represents    the  area    included  between 
J* 

the  curve.  AB  and  its  projection  upon  the  </>-axis. 
Hence  the  heat  necessary  to  produce  any  reversible  change 
is  represented  by  the  area  under  the  curve  in  the  T$-plane. 
The  differential  form  of  equation  (1)  leads  at  once 
to  the  expression 


At  any  point  n  of  the  curve  AB  draw  the  tangent 
nm  and  construct  the  infinitesimal  triangle  dtd<j>. 
Then  from  similar  triangles 


b  :dT, 

i.e.,  if  from  any  point  in  a  curve  representing  a  re- 
versible process  a  tangent  be  drawn,  the  length  of  the 
subtangent  on  the  <£-axis  represents  the  momentary 
value  of  the  specific  heat  for  that  change. 

In  Fig.  2  (A}  let  AmBnA  represent  a  reversible  cycle 
in  which  AmB  is  the  forward  stroke  and  the  area 
aAniBb  represents  the  heat  received  from  external 
sources,  and  where  BnA  is  the  return  stroke  and  the 
area  aAnBb  represents  the  heat  rejected.  On  the 
completion  of  the  cycle  the  intrinsic  energy  has  regained 
its  initial  value  and  therefore  the  difference  between 


CARNOT  CYCLE. 


the  heat  received  and  the  heat  rejected,  i.e.,  the  magni- 
tude of  the  enclosed  area  AmBnA,  must  represent  the 
amount  of  heat  changed  into  work. 

Carnot  Cycle. — In  the  case  of  the  Carnot  engine  this 
choice  of  coordinates  leads  to  a  beautiful  simplicity. 
The  cycle  Fig.  2  (J5)  becomes  a  rectangle  consisting 


(A) 


FIG.  2. 


(B) 


of  (1)  the  isothermal  expansion  cd,  during  which  is 
received  the  heat  Ql  =  i  Tl  d<f>  =  T1(<f>t-$l),  repre- 

J 4>\ 

sented  by  the  total  rectangle  facdfa',  (2)  the  adiabatic 
line  de\  (3)  the  isothermal  compression  during  which 

is  rejected  the  heat  Q2=   I    2T2d<f>  =  T2((f>2-(}>l),  repre- 
*/  <f>i 

sented  by  <£i/e<£2J  and  (4)  the  adiabatic  compres- 
sion fc.  The  heat  changed  into  work  is  Qt—  Q2  = 
771(^-</>i)-^(^-^)  =  (7t1-7T2)(^-^).  The  effi- 
ciency, T)  =  —^ — -,  is  simply  the  ratio  of  the  rectangles 

Vi 

cdef  and  ^cd^,  and  as  these  have  the  same  base,  the 


6 


THE   TEMPERATURE-ENTROPY  DIAGRAM. 


areas  are  proportional   to   the  altitudes  and  at  once 

T  —T 

the  efficiency  becomes  >j  =  ~^m — "• 
•*  i 

Isodiabatic     Cycles.  —  In     deducing     the     equation 
d<f>  =  -jjf-  for  reversible  cycles,  use  is  made  of  the  Carnot 

cycle  and  it  is  shown  that  no  other  cycle  can  have  a 
greater  efficiency.  There  are,  however,  a  number  of 
other  cycles  having  the  same  efficiency  as  the  Carnot 
within  the  same  range  of  temperature. 

In  Fig.  3,  let  the  reversible  cycle  abed  be  formed  by 


FIG.  3. 

the  two  isothermals  ab  and  cd  and  of  the  two  curves 
fee  and  da.  The  curve  be  is  arbitrary,  but  da  is  drawn 
like  it,  being  simply  displaced  to  the  left.  The  heat 
turned  into  work  during  the  cycle  is  abed,  or  equal  to 
that  of  the  equivalent  Carnot  cycle  abef,  but  the  heat 
supplied  is  d'daa'  +afaW,  or  is  greater  than  that  needed 
to  perform  the  same  work  with  the  Carnot  cycle  by 
the  amount  d'daa'.  Now  the  heat  rejected  from  b  to  d 
consists  of  the  two  parts,  cdd'cf,  which  passes  to  the 


ISODIABATIC  CYCLES.  7 

refrigerator  at  the  lowest  temperature  T2  and  is  of 
necessity  lost,  and  bb'c'c,  which  is  rejected  during  a 
dropping  temperature  and  which  is  strictly  equal  to 
the  heat  required  to  carry  out  the  reverse  operation  da. 
Instead  of  being  wasted,  the  heat  rejected  along  be 
might  be  stored  up  and  returned  to  the  fluid  along  da, 
thus  making  no  further  demands  upon  the  source  of 
heat.  In  this  manner  the  one  operation  would  balance 
the  other  and  the  heat  actually  required  from  an 
external  source  would  be  represented  by  afaW,  as  in 
the  Carnot  cycle.  The  heat  actually  rejected  would 
also  be  represented  by  d'dcc',  equal  to  a'feb',  as  in  the 
Carnot  cycle.  Thus  the  adiabatic  lines  of  the  Carnot 
cycle  would  be  replaced  by  two  lines  be  and  da,  along 
which  the  interchanges  of  heat  are  compensated.  The 
efficiency  of  such  isodiabatic  cycles  would  thus  be  equal 
to  that  of  the  Carnot  cycle. 

The  operations  along  the  lines  be  and  da  may  be 
imagined  as  follows.  Heat  only  flows  spontaneously 
from  one  body  to  another  at  a  lower  temperature. 
Thus  the  heat  given  up  along  be  can  be  stored  and 
utilized  to  effect  the  operation  da,  if  the  process  be 
subdivided  into  very  small  differences  of  temperature 
and  each  portion  of  the  heat  rejected  at  the  momentary 
temperature.  Thus  the  heat  rejected  along  be  between 
the  temperatures  T  +dT  and  T  (Fig.  3)  is  returned 
along  a  portion  of  da  at  the  same  temperature. 

The  difficulty  is  to  find  a  practical  regenerator  to 


8  THE  TEMPERATURE-ENTROPY  DIAGRAM. 

perform  this  duty.  It  must  have  large  heat-conducting 
surfaces,  and  consist  of  a  number  of  subdivisions  at 
all  temperatures  between  T±  and  T2,  and  there  must 
be  no  conduction  of  heat  from  any  part  to  the  next  at 
lower  temperature.  To  accomplish  the  operation  along 
be,  the  working  fluid  passes  successively  through  these 
divisions  and  deposits  in  each  a  part  of  its  heat.  Dur- 
ing the  process  da  the  fluid  passes  again  through  the 
divisions  but  in  a  reverse  direction,  from  the  coldest 
to  the  hottest.  This  ideal  process  can  only  be  roughly 
realized  as  the  regenerator  has  a  limited  conductibility, 
permits  the  flow  of  heat  between  adjacent  sections, 
and  can  only  store  up  heat  if  the  fluid  is  much  hotter 
and  refund  it  if  the  fluid  is  much  colder  than  itself. 
Therefore  only  the  upper  portion  of  the  line  be  can 
actually  be  utilized  to  refund  heat  gratuitously  along 
the  lower  portion  of  da.  Hence  in  practice  it  should 
be  possible  to  obtain  a  higher  efficiency  with  an  engine 
working  on  the  Carnot  cycle,  because  those  working 
on  the  isodiabatic  cycle  have  the  added  losses  of  the 
regenerator.  But  theoretical  efficiency  is  only  one 
of  several  factors  entering  into  the  design  of  an  actual 
engine  and  may  be  more  than  balanced  by  the  increased 
size,  cost,  strength,  etc.,  required  to  attain  it. 

A  different  type  of  regenerator  is  described  by  Swin- 
burne. The  engine  consists  of  a  primary  cylinder  to 
perform  the  external  work  and  a  secondary  one  to  act 
as  a  regenerator.  Each  cylinder  possesses  sides  and 


ISODIABATIC    CYCLES.  9 

piston  absolutely  impermeable  to  heat,  but  an  end 
which  is  a  perfect  heat  conductor.  The  cycle  of  the 
working  fluid  in  the  primary  cylinder  consists  of  two 
isothermal  and  two  constant-volume  processes.  Placed 
'in  contact  with  a  source  of  heat  of  constant  tempera- 
ture TI  the  fluid  expands  isothermally  up  to  the  end 
of  the  stroke;  then  removed  from  the  source  of  heat, 
heat  is  extracted  at  constant  volume  until  the  tem- 
perature has  fallen  to  T2',  next  placed  in  contact  with 
a  refrigerator  of  constant  temperature  T2  the  working 
fluid  is  compressed  isothermally  until  the  initial  volume 
is  regained.  The  second  cylinder  operates  only  during 
the  constant  volume  changes.  With  its  working  sub- 
stance compressed  into  the  clearance  space  and  at  the 
temperature  TI  its  conducting  end  is  brought  into 
contact  with  the  corresponding  end  of  the  primary 
cylinder,  just  as  the  latter  is  removed  from  the  source 
of  heat.  The  two  charges  in  their  respective  cylinders 
.are  thus  maintained  separate  while  the  temperature  is 
always  equalized.  The  piston  in  the  secondary  cylin- 
der now  moves  forward,  performing  external  work  at 
the  expense  of  the  internal  energy  of  its  own  charge 
and  of  the  heat  received  from  the  primary  cylinder, 
cv(Ti—T2}.  This  process  is  to  occur  so  slowly  that 
only  an  infinitesimal  drop  of  temperature  exists  through- 
out the  combined  masses.  When  not  in  contact  with 
the  main  cylinder  the  auxiliary  remains  inactive.  For 
its  return  stroke  the  secondary  is  placed  in  contact  with 


10        THE    TEMPERATURE-ENTROPY   DIAGRAM. 

the  primary  as  the  latter  is  removed  from  the  refrig- 
erator, and  external  work  of  an  amount  equal  to  that 
developed  during  the  forward  stroke  must  be  performed 
upon  it,  thus  gradually  restoring  its  own  temperature 
and  that  of  the  primary  by  returning  to  it  the  heat* 
c*(Ti-T$. 

Thus  while  the  entropy  of  the  working  charge  is 
decreasing  along  be  (Fig.  4)  that  of  the  auxiliary  charge 
is  increasing  an  equal  amount  along  be',  and  while  that 
of  the  auxiliary  charge  decreases  along  d'a  (curve  c'b 


a                        b 

r 

\        I 

\ 
\ 

/ 

\    / 

\ 

d 

d' 

c 

1 

FIG.  4. 

moved  to  the  left)  the  entropy  of  the  working  charge 
undergoes  an  equal  increase  at  the  same  temperatures. 
Thus  during  the  .isodiabatic  processes  the  combined 
entropies  of  the  two  charges  possess  a  constant  value. 
Decrease  in  Efficiency  due  to  Irreversibility. — In  all 
actual  engines  where  the  charge  enters  and  leaves  the 
cylinder  each  cycle,  a  small  difference  of  pressure  and 
hence  of  temperature  must  always  exist  between  the 
fluid  in  the  supply-pipe  and  in  the  cylinder,  and  between 
that  in  the  cylinder  and  in  the  exhaust-pipe.  Thus, 
let  TI  and  T2  (Fig.  5)  represent  the  temperatures  of 


IRREVERSIBLE  CYCLES. 


11 


source  and  refrigerator;  then,  for  a  Carnot  cycle  abed 
represents  the  heat  utilized,  while  dcgh  represents  that 
rejected.  If  it  be  assumed  that  no  loss  of  heat  is 
experienced  by  the  working  fluid  during  admission, 
but  simply  the  drop  in  temperature  7\7Y,  then  area 
afb'g'li  must  equal  area  dbgh.  Similarly,  the  heat 
rejected  at  7Y  from  the  actual  engine  (area  d'c'g'h) 
would  at  the  temperature  T2  be  equal  to  the  area  dee'h, 
and  thus  exceeds  that  rejected  in  the  ideal  case  by  the 


a         Supply  Pipe           b 

Ci'              Cylinder 

I,' 

d             Cylinder 

r' 

5T 

(I            Exhaust  Pipe       C 

99' 


FIG.  5. 


area  cee'g.    The  efficiency  of  the  actual  cycle  is  there- 
fore reduced  by  these  irreversible  processes. 

It  is  important  to  notice  that  the  increased  exhaust 
subdivides  into  the  two  areas  cc\gfg  and  ci'ee'g',  which 
represent  the  losses  incurred  during  admission  and 
exhaust  respectively.  Thus  if  there  were  no  throttling 
during  exhaust  the  adiabatic  expansion  would  be  from 
b'  to  C/,  and  therefore  cci'g'g  must  represent  the  loss 
due  to  throttling  at  admission  alone.  The  loss  is  in 
each  case  equal  to  the  increase  in  entropy  during  the 


12         THE   TEMPERATURE-ENTROPY   DIAGRAM. 

throttling,  multiplied  by  the  lowest  available  tem- 
perature. Thus  it  follows  that  any  irreversibility, 
which  is  always  accompanied  by  a  growth  of  entropy, 
must  cause  a  decrease  in  the  thermal  efficiency  of  a 
cycle.  Of  course,  the  net  efficiency  is  always  still  fur- 
ther reduced  by  actual  heat  losses  due  to  conduction 
and  radiation. 

Therefore  in  working  between  any  two  temperatures  the 
highest  possible  efficiency  will  not  be  attained  unless  all 
the  heat  received  is  taken  in  at  the  upper  temperature, 
and  all  the  heat  rejected  is  given  out  at  the  lower  tem- 
perature. The  only  exception  to  this  is  in  the  case 
of  the  isodiabatic  cycles  already  considered. 


CHAPTER  II. 

THE    TEMPERATURE-ENTROPY    DIAGRAM    FOR 
PERFECT  GASES. 

A  PERFECT  gas  is  defined  as  a  substance  whose  specific 
volume,  pressure,  and  temperature  satisfy  the  equation 

pv  =RT, 

and  whose  specific  heat  at  constant  pressure,  cp,  and 
at  constant  volume,  cv,  are  constants. 
Starting  from  the  equation  for  the  conservation  of 

energy, 

dQ=A(dE+pdv), (1) 

for  a  change  of  condition  at  constant  volume  we  obtain 
dQ  =  AdE=cvdT; 

s* 

whence  E=-jT+ constant (2) 

The  internal  energy  of  a  perfect  gas  is  thus  seen  to 
depend  solely  upon  the  absolute  temperature,  so  that 
an  isothermal  process  is  also  an  isodynamic  one. 

Let  us  consider  next  a  unit  weight  of  gas  in  the 
condition  (pvT),  and  let  the  temperature  increase  from 
T  to  T\,  once  at  constant  volume  and  a  second  time 

at  constant  pressure.    In  the  first  case  the  condition 

13 


14         THE  TEMPERATURE-ENTROPY  DIAGRAM. 

is  changed  from  (pvT)  to  (p\vT^)  by  the  addition  of 
the  heat,  cv(Ti—  T);  in  the  second  case  from  (pvT) 
to  (pViTi)  by  the  addition  of  the  heat  cp(Tl-T\ 

Since  the  change  of  temperature  is  the  same  in  both 
cases  the  increase  in  internal  energy  is  the  same  and 
therefore  the  difference  between  the  heat  added  in  the 
two  cases  must  represent  the  difference  in  the  external 
work  performed.  Then  we  have 


whence  cp—  cv=AR  =  cv(k—  1).      .    .     (3) 

Substituting,  this  relation  in  the  expression  for  in- 
ternal energy  gives 


The  entropy  for  any  condition  can  now  be  determined 
without  difficulty.     Since 


dv 


r  r          A  /•     v 

=  J  ^=cvj  ^r  +  AJ  pyr  +  constant, 


+  AR  I  —  [-constant. 
J  v 


pv 
~R 

-cv)  loge  v+constant, 


cv  loge  p  +  ~  loge  v   +  constant, 

Cv  loge  pvk  +  constant  i     ......     (5a) 


PERFECT  GASES. 


15 


By  use  of  the  characteristic  equation  pv  =  RT,  either 
p  or  v  can  be  eliminated  so  that  the  entropy  may  also 
be  expressed  as 

.    .    (56) 


i-fc 


+  constant3    .     .     .     (5c) 


THE  fundamental  heat  equations  for  a  perfect  gas  are 


.     .     .    .     (6) 


and  cp-cv  =  AR, (7) 

which  for  reversible  processes  give  three  different  expres- 
sions for  entropy : 


•     •     (8) 


The  curves  for  constant  volume  and  constant  pressure, 
respectively,  in  the  T^-diagram  become 

T, 
$2-$i  =  cvloge7jr (9) 


16          THE  TEMPERATURE-ENTROPY  DIACRAM. 

and  &-&-**«.  a?  ......  (10) 

L  i 

If  p  =  f(v)  be  the  pv-projection  of  any  curve  on  the 
characteristic  surface  pv  =  RT.',  to  find  the  T^-projec- 
tion  of  the  same  curve  it  is  only  necessary  to  find 

—  =fr(T)  or  ^=/2(T)  from  the  above  equations  and  to 

Vi  pi 

substitute  them  in  the  first  and  second  forms,  respec- 
tively, of  equations  (8). 

The  most  general  form  of  p  =  f(v)  with  which  the 
engineer  is  concerned  is  that  of  an  indicator  card, 

namely, 

pvn  =  pjV^  =  a  constant,    .     .     .     .  (11) 

where  the  exponent  is  determined  from  any  two  points 
by  means  of  the  formula 


_ 
log  va-log< 

The  respective  projections  of  equation  (11)  on  the 
TV-  and  Tp-planes,  as  found  by  combination  with  the 
characteristic  equation,  are 

Tlvln-l  =  T^n~l  =  a  constant    .     .     .  (12) 

and  fj^j  *  -Tap,  »  —a  constant,       .    .  (13) 

whence  it  follows  that 


PERFECT  GASES.  17 

The  substitution  of  equations  (14)  in  the  corresponding 
forms  of  equations  (8)  gives 


<f>2-<t>l  =  Cp  loge    T  -  (C,  -Cj  log,        i 


Equations  (11),  (12),  (13),  and  (15)  represent  simply 
different  projections  of  the  same  curve  on  the  charac- 
teristic surface  upon  different  planes. 

From  equation  (2),  p.  4,  for  perfect  gases, 

I/TF  m 

d<f>  =  c,    or    #,-^-clog.r.  .    .     (16) 


Comparison  of  equations  (15)  and  (16)  shows  that  the 
specific  heat  for  the  general  expansion  plvln  =  p2v2n  is 

'=<W      .....     (17) 

i.e.,  the  specific  heat  of  any  expansion  of  a  perfect  gas, 
ptfn  =  constant,  can  be  expressed  as  a  function  of  the 
specific  heat  at  constant  volume,  the  ratio  of  cp  and  cv,  and 
the  exponent  n  of  the  expansion. 


18         THE  TEMPERATURE-ENTROPY   DIAGRAM. 

Let  us  consider  equations  (11),  (15),  and  (17)  for 
the  following  special  cases: 

(1)  For  T  =  const.,  pv  =  R T  =  constant,  and  n  =  l. 

Equation    (17)  becomes  c  =  cv-  •-.._-.    =  °°;   i-e.,   for 

an  isothermal  change  the  heat  capacity  of  the  substance 
is  infinite;  that  is  to  say,  no  matter  how  much  heat  is 
added  the  temperature  will  not  change. 

Equation  (15)  becomes  </>2  —  (f>1=<xi-Q;  that  is,  the 
value  of  <£  may  undergo  any  change  whatever  and  the 
TV-curve  simply  becomes  T  =  const. 

(2)  For    <f>  =  constant,    equation    (15)    gives    n  =  k, 

£ ft, 

and  equation  (17)  becomes  c  =  cv-, — 7  =0;     i.e.,  for  an 

K  —  1 

isentropic  change  the  heat  capacity  of  the  substance 
is  nil;  that  is  to  say,  the  temperature  of  the  substance 
can  be  increased  or  diminished  without  the  addition 
or  subtraction  of  heat  as  heat.  Equation  (11)  becomes 
pvk= constant. 

(3)  For  p  =  constant,  equation  (11)  becomes  vf^vf, 
hence  n  equals  zero. 

0  —  k         c 
From  (14)  c  =  cv^ — T-=CV-  —  =  CP,  and  equation  (15) 

Cv 

T 

becomes  <£2-<£i  =  c*>  ioge^r,  as  already  found  in  equa- 
tion (10). 

(4)  For  v  =  constant,  the  only  value  of  n  which  will 
satisfy    pAn  =  p2V   is  n=oo;    so  that  equation  (11) 
becomes  v= constant. 


PERFECT  GASES. 


19 


Equation   (17)  gives  c  = 


—  —  r-  =  cv,  whence  from 
T 


\oge~,  as  previously  found  in 
-*  i 


equation  (15)  <f>2  —  <j>l 

equation  (9). 

We  are  now  in  a  position  to  transfer  any  curve  from 
the  pv-plane  to  the  7^-plane  as  soon  as  we  know  the 
value  of  cv  for  the  given  gas. 

In  Fig.  6  let  abed  represent  a  cycle  consisting  of  two 


curves  of  constant  volume  and  two  of  constant  pres- 
sure indicated  by  a  rectangle  in  the  yw-plane.  In  the 
T^-plane  start  with  the  value  of  the  entropy  at  a  as  the 
zero-point.  The  curve  ab  will  be  of  the  nature  shown, 
becoming  steeper  as  T  increases  because  the  subtangent 
at  any  point  represents  the  value  of  cv  and  this  is  a 
constant  for  perfect  gases.  Arriving  at  6,  the  curve 
of  constant  pressure  will  assume  some  such  position  as 
shown,  be  will  not  be  as  steep  as  ab  at  the  point  of 
intersection  because  cp>cv,  i.e.  the  subtangent  of  be 


20 


THE  TEMPERATURE-ENTROPY   DIAGRAM. 


is  greater  than  the  subtangent  of  ab.    Two  similar 
curves  cd  and  da  complete  the  cycle. 
All  possible  variations  of  the  curves  pvn  =  constant, 

n  —  k         T 

or  02-0i  =  cI, r-loge^r,  are  summarized  in  the  fol- 

n  —  1          1 1 

lowing  table  and  diagrams: 


pv-coordinates. 


T^-coordinates. 


n 

Form  of  the  Curve. 

*'£i* 

Form  of  the  Curve. 

0 

p  =  constant 

CP 

1  T    and    <j>   increaje    or   de- 

0<n<l 

pt)i  =  constant 

>cp 

1      crease  together 

1 
Kn<fc 

fc 

pv  =  constant  (isothermal) 
pvn  =  constant 
pifc  =  constant  (isentropic) 

j  Nega- 
I   tive 
0 

T  =  constant  * 
j  T  increasing  and  <£  decr'sing 
1  ^'decreasing  and  <j>  incr'sing 
<j>  =  constant 

fc<n<oo 

pvn  =  constant 

<ca 

j  T    and    <j>    increase    or   de- 

00 

v=  constant 

cv 

1      crease  together 

*  During  an  isothermal  change  of  a  perfect  gas  all  the  heat  added  performs 
external  work,  hence  in  the  diagrams  the  isodynamic  lines  are  superimposed 
upon  the  isothermals. 


|<n<fc 


\<n<k 


0          O 

Fia.  7. 


PERFECT  GASES.  21 

The  7^-Projection  of  any  pu-Curve.  —  In  case  the 
curve  p--=f(v)  does  not  possess  an  equation  of  the 
simple  form  pvn  =  c,  and  it  thus  becomes  more  difficult 
to  determine  the  corresponding  expression  <j>=fi(T), 
it  will  sometimes  prove  simpler  to  avoid  the  analytical 
solution  and  to  find  the  desired  curve  graphically  by 
plotting  it  point  by  point  at  the  intersections  of  the 
proper  constant  pressure  and  constant  volume  curves. 

This  may  be  done  the  more  readily  since  the  curves 
of  constant  pressure  and  constant  volume 


and    0=cvloge  T+c2 


are  of  such  a  character  that  the  intercepts  upon  iso- 
thermals  made  by  any  two  constant  pressure  or  con- 
stant volume  curves  are  of  equal  lengths  and  equal 
respectively  to 

4<f>=ARloge-    and     40  =  A/Hog*  ^ 

Hence  to  draw  a  series  of  constant  pressure  or  of 
constant  volume  curves  in  the  T^-plane  the  following 
procedure  is  necessary  and  sufficient: 

(1)  Determine  accurately  the  contour  of  the  two 
curves 

<f>=Cp  logeT    and    <£  =  cv  loge  T7. 


(2)  Construct  a  template  of  both  curves. 


22 


THE    TEMPERATURE-ENTROPY  DIAGRAM. 


(3)  Determine 


A$  =  AR  loge  —     and 


P* 


for  the  necessary  pressures  and  volumes. 

(4)  Along    the    isothermal    passing    through    pivi} 
lay  off  the  values  of  J^  determined  in  (3). 

(5)  Using  the  templates  draw  through  these  points 
the  corresponding  pressure  or  volume  curves. 

Having  these  base  lines  once  constructed,  any  irregular 
curve,  p=f(v),  may  be  at  once  transferred  from  the 
pv-  to  the  !T^-plane  by  noting  the  values  of  pressure 
and  volume  at  a  sufficient  number  of  points  to  give  a 
smooth  curve. 

Relative  Simplicity  of  the  T(j>-  and  pv  -Projections. — 
In  graphical  work  where  it  may  be  necessary  to  plot 
isothermals  and  reversible  adiabatics  as  well  as  con- 
stant pressure  and  constant  volume  curves,  it  is  simpler 
to  use  the  T<j>-  than  the  pu-plane.  This  is  at  once 
evident  from  the  following  tabulation: 


Curves. 

pv-plane. 

T^-plane. 

Constant  pressure  

Horizontal  lines.  .  .  . 
Vertical  lines    

<  <£  =  eploge  T  +  CI 
'  The  curves  are  all  alike 
(  <t>  =  cvloge  7'  +  c2 

"        temperature  .  . 
**        entropy  

I  pv  =  c 
\  No  two  curves  alike 
{  vvn=c 
*  No  two  curves  alike 

<  The  curves  are  all  alike 
}  Horizontal  lines 

}  Vertical  lines 

The  Logarithmic  Curve  y  =log  x. — The  logarithmic 
curve  forms  the  basis  not  only  of  the  graphical  method 


PERFECT  GASES. 


23 


for  plotting  any  polytropic  curve  in  the  pv-plane,  but 
also  of  the  method  for  projecting  any  curve  from  the 
TV-  and  pu-planes  into  the  T(£-plane.  Such  a  curve 
can  be  constructed  very  quickly  from  a  logarithmic 
table,  and  in  case  much  of  this  work  must  be  done  a 


FIG.  8. 

template  can  be  constructed  and  the  curve  thus  be 
drawn  whenever  needed. 

In  case  logarithmic  tables  are  not  at  hand  and  even 
if  they  are,  and  some  arbitrary  value  of  x  is  assumed 
as  unity,  the  following  graphical  method  for  construct- 
ing the  logarithmic  curve  is  often  of  great  convenience. 


24         THE    TEMPERATURE-ENTROPY  DIAGRAM. 

Let  OxQ  (Fig.  8)  represent  the  arbitrary  unit  of  x. 
Through  0  draw  any  straight  line  OS.  Then  starting 
with  0  as  a  centre  and  OXo  as  the  first  radius,  con- 
struct the  series  of  arcs  and  perpendiculars,  determining 
the  series  of  values, 


From  the  series  of  similar  triangles  these  quantities 
are  related  as  follows: 


_____.___ 

m~Xn-l~        '~~X3~X2~Xi~XQ~Xi~X2~X3~'         ~    X 

whence  Xi  =  mr1x  Xi  =  mx0 


Xn  =  mXn-  1  = 


Taking  the  logarithms  and  remembering  that  log 
=0  and  setting  log  m  =  D,  we  obtain 


logm~nx0=--n-logm=  -n-D  =  log  — 


log x3  =\ogm~sXQ^—  3  log  m=  -3-D  =  log  — 

XQ 

\ogx2  =logw~2xo=-2  logw--2  Z)  =  log  — 

XQ 


PERFECT   GASES.  25 


log  x\  =  log  m~-lx0  =  —  1  log  m  =  —  1  D  =  log  — 

XQ 

*        =         0  =  log  — 

XQ 
v 

=+1  logw=+l  D  =  log  — 

XQ 
•jr 

=+2-logm=+2  D  =  log  — 

XQ 
v 

=+3  logw=+3  D  =  log- 

XQ 


Xn 

logXn=logmnXo  =  +n  logm  =  +n  D=log  — 

XQ 

This  method  of  construction  has  thus  given  a  series 
of  points  x  such  that  the  logarithms  of  any  two  succes- 
sive points  of  the  series  differ  by  a  common  amount  D. 
Therefore  to  construct  the  desired  logarithmic  curve 
y=\ogx,  starting  at  XQ,  lay  off  on  the  vertical  lines 
through  Xi;  X2,  X3,  . . .  Xn,  and  Xi,  x2,  x3,  . . .  xn  the 
distances  D,  2D,  3D,  ...nD  and  -D,  -2D,  -3D, 
...  —  nD,  respectively,  and  connect  them  with  a  smooth 
curve. 

In  case  any  multiple  of  this  curve  is  desired,  such  as 

T 
y—a-logx,  as,  for  example,    4(}>=cv  loge  -^  ,  the  ordi- 

*  o 

nates  of  the  curve  y  =  log  x  may  each  be  multiplied  by  a 
and  the  product  aD,  2aZ>,  . .  .  ,  laid  off  as  before.    A 


26 


THE    TEMPERATURE-ENTROPY    DIAGRAM. 


simple  and  more  convenient  method  is  that  shown  in 
Fig.  9. 

Lay  off  from  0  the  distance  unity  (x0  =  l),  from  x0 
drop  a  perpendicular  equal  to  a  units.  Draw  OA. 
Then  from  similar  triangles  if  OB  equal  log  x,  BC 
will  equal  a-\ogx.  Therefore  to  determine  a\ogx  =  y 
for  any  value  of  x,  take  log  x  from  the  log-curve  with 


FIG.  9. 

the  dividers  and  locate  B.  Then  measure  BC  with 
the  dividers. 

Graphical  Construction  of  the  Polytropic  Curves 
(pvn  =  c)  in  the  pv-Plane. — To  construct  the  polytropic 
curves  choose  any  convenient  volume,  as  OA  (Fig.  10), 
for  unit  length,  and  then  through  this  point  draw  in 
any  convenient  manner  the  logarithmic  curve  p  =  logv. 

From  the  origin  lay  off  to  the  left  OA'  =  OA,  and 
then  A'B  =—n.  Draw  the  straight  line  OB,  and  the 
constant-volume  curve  through  A.  Choose  any  number 
of  points  on  this  line,  as  1,  2,  3,  . .  . ,  according  to  the 
number  of  curves  desired,  and  connect  each  point  by  a 
straight  line  to  the  origin. 


PERFECT    GASES. 


27 


To  determine  the  pressures  on  the  curves  1, 2,  3,  . . . , 
corresponding  to  any  volume,  as  v\,  take  log^i  from 
the  log-curve  and  lay  off  at  the  left  of  0,  as  Ovi.  Drop 


a  vertical  line  through  vi  until  it  intersects  OB  at  a\. 
Project  horizontally  on  the  log-curve  at  61.  Through  61 
erect  a  perpendicular  intersecting  01,  02,  03,  .  .  .  ,  at 
hi,  h2,  ha,  . . . ,  respectively.  Draw  horizontal  lines 


28        THE    TEMPERATURE-ENTROPY   DIAGRAM. 

through  these  points  intersecting  v\  at  pif  p2,  ps,  .  .  .  . 
These  are  the  desired  points. 

The  proof  of  this  is  simple.     From  similar  triangles, 

»i'oi  :  -log  »i  --»:-!, 

Vi'a  =  —  n  log  vi  =  log  v  i~n  =  W. 


Again  from  similar  triangles, 

bhf.Ob    =  A1:OA, 


To  continue  the  curves  to  the  left  of  A,  prolong  BO 
into  pv-plane,  as  shown,  and  lay  off  log  v  to  the  right 
of  0.  Then  proceed  as  before. 

This  same  method  can  be  used  to  plot  the  TV-pro- 
jection of  the  polytrope,  Tvn~1  =  c,  provided  the  auxiliary 
line  OB  is  determined  by  the  coordinates  [  —  1,  —  (n—  1)] 
instead  of  (  —  1,  —  n). 

This  construction  may  be  used  backwards  to  deter- 
mine n  for  a  given  polytropic  curve,  or  if  the  curve  is 
only  approximately  polytropic  a  mean  value  of  n  can 
be  determined. 

Construction  of  the  Isothermals,  pv=c,  in  the  pv- 
Plane.  —  Draw  the  curves  pi=c  and  Vi  =c,  which  bound 
that  portion  of  the  pu-plane  (Fig.  11)  in  which  the 
isothermals  are  to  be  constructed.  To  construct  the 
isothermal  through  any  point,  as  A,  draw  the  curve 


PERFECT  GASES. 


29 


v  =va.  Draw  the  straight  lines  01,  OH,  OIII, . . . ,  which 
intersect  v  =va  in  points  1,  2,  3,  4,  ....  Draw  a  series 
of  vertical  lines  through  7,  II,  HI,  .  . .  ,  and  horizontal 
lines  through  1,  2,  3, . . . ,  then  the  points  of  intersection 
II,  112,  1 1 13, .  .  .  ,  are  points  on  the  desired  isothermal. 

I        IT  Til  IV  V  VI      VII 


FIG.  11. 

The  isothermals  through  the  successive  points  7,  77, 
777,  . . .  ,  can  be  drawn  by  using  the  same  radial  and 
vertical  lines.  The  only  extra  construction  necessary 
being  the  horizontal  lines  through  the  points  of  intersec- 
tion of  7,  77,  777,  .  . .  ,  with  the  radial  lines. 

This  construction  is  very  simple  if  carried  out  on 
plotting  paper,  as  then  only  the  radial  lines  need  to  be 
constructed. 


30          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  justification  for  this  construction  5s  found  in  the 
equation  of  the  curve  itself.  Thus  pv=c  may  be 
construed  graphically  to  mean  that  the  area  of  the 
rectangle  included  between  the  axes  of  p  and  v  and 
the  constant  pressure  and  constant  volume  lines  through 
any  point  on  the  curve  is  a  constant.  Thus  take  the 
points  (A)  and  (2,11)  on  the  isothermal  through  A. 
If  these  points  lie  on  the  same  isothermal  rectangle  OA 
must  equal  rectangle  0  2,  II.  Now  A  OHp  =  A  Oil  1 1' 
and  A  022'  =  A  02va  and  A  21 1 A  =  A  2/7(2, 77),  whence 
rectangle  2' A  =  rectangle  va(2, 77)  and  therefore 
rectangle  OA  =  rectangle  0(2,  77). 

Slide  Rule  Construction  of  Isothermals. — If,  in  place 
of  drawing  isothermals  through  points  already  known, 
it  becomes  necessary  to  determine  them  for  definite 
temperatures  the  work  of  plotting  may  be  carried  on 
directly  from  the  slide  rule.  Thus  set  the  slider  to 
mark  the  product  R  T,  and  then  the  volumes  corre- 
sponding to  any  desired  number  of  pressures  may  be 
read  from  one  setting  for  each  value.  The  slider  must 
be  reset  for  each  new  isothermal. 

Representation  of  W,  Q,  and  AE  in  the  pv-  and 
T<£-Planes. — The  pv-plane  gives  at  once  the  external 
work  performed  during,  and  the  77^-plane  the  heat 
interchange  involved  in,  any  reversible  process.  To 
represent  the  change  of  internal  energy  two  methods 
are  available  in  both  planes. 

(1)  Since  isothermal  and  isodynamic  lines  are  coro- 


PERFECT    GASES.  31 

cident  it  is  only  necessary  to  determine  JE  per  unit 
increase  of  temperature  and  then,  by  assuming  some 
arbitrary  zero  of  internal  energy,  to  assign  values  to 

the    different    isodynamics.     Thus     4E=-1-^-  =  -,  —  —  » 

A;—  1      A;  —  1 

and  AE  per  unit  change  of  temperature  equals 
r  —  7  =  ~A-  Hence  the  change  of  intrinsic  energy  during 

K  —  1      A. 

any  process  may  be  found  by  noting  tha  initial  and 
final  state  points  with  reference  to  the  isodynamic 
lines. 

(2)  If  a  substance  expands  adiabatically,  performing 
work  at  the  expense  of  its  internal  energy,  and  if  this 
process  occurs  so  slowly  as  to  prevent  increase  of 
kinetic  energy  and  to  permit  the  maintenance  of  a 
uniform  temperature,  and  if  there  be  neither  internal 
nor  surface  friction,  then  the  expansion  will  at  the 
same  time  be  isentropic.  For  such  an  isentropic 
expansion  the  area  under  the  curve  in  the  pv-plane  will 
not  only  represent  the  external  work  performed  but 
will  at  the  same  time  be  a  measure  of  the  decrease  of 
internal  energy.  Thus  between  any  two  points  a  and 
b  upon  the  same  isentrope  there  must  exist  the  difference 
of  internal  energy, 


where  n  equals  the  exponent  of  the  adiabatic  curve. 


32        THE  TEMPERATURE-ENTROPY  DIAGRAM. 
If  the  point  b  be  removed  to  infinity  there  results 

v      F       PaVa  - 

Ilia  —  tii^  =  ---  T» 

00     n-l' 

i.e.,  the  decrease  in  internal  energy  during  frictionless 
adiabatic  expansion  from  any  finite  condition  a  to 

infinity  is  represented  by        °.     This,  however,  is  not 
n  —  1 

the  total  value  of  the  internal  energy  at  condition  a, 
because  by  definition, 

Internal  energy  =  vibration    energy    plus    disgregation 

energy; 
=  kinetic  energy  plus  potential  energy; 

or  E=S+I, 

and  at  infinity  while  the  vibration  or  kinetic  energy 
has  become  zero  with  the  disappearance  of  both  pressure 
and  temperature,  the  disgregation  or  potential  energy 
has  assumed  its  maximum  value  due  to  infinite  increase 
in  volume.  Hence  #00=£,X).+/00=0  +  /00=/00,  and  we 
may  write 


Thus  the  internal  energy  of  a  perfect  gas  can  be  deter- 
mined up  to  the  value  of  a  certain  unknown  constant, 

/.- 

In  the  case  of  a  perfect  gas  the  molecules  are  sup- 
posed to  be  so  far  apart  as  to  be  beyond  the  sphere  of 


PERFECT  GASES. 


33 


mutual  attraction,  so  that  the  value  of  Ix  is  already 
established  at  finite  distances,  and  changes  in  internal 
energy  are  directly  proportional  to  changes  in  tem- 
perature. 

(a)  To  find  the  difference  in  internal  energy  between 
any  two  state  points  a  and  6  of  a  perfect  gas  we  have 


W  _  F  a-  T    _     aa 

~k-l+1«    k-1 


- 

*.-!    k-1 


or 


A  (Eb  - 


cvTb-  cvTa. 


Interpreted  graphically  (Fig.  12)  this  states  that  the 
difference   between  the   intrinsic   energy   of  any  two 


state  points  is  represented  in  the  pu-plane  by  the  dif- 
ference of  the  areas  under  the  adiabatics  drawn  through 
the  respective  points  and  extended  to  infinity;  or  the 


34        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

heat  equivalent  of  the  difference  in  kinetic  energy  is 
represented  in  the  7^-plane  by  the  difference  of  the 
areas  under  the  constant  volume  curves  drawn  through 
the  respective  points  and  extended  to  infinity. 

Here  again,  although  both  diagrams  are  infinite,  the 
T(j>  is  the  easier  to  construct  accurately. 

(6)  To  avoid  the  use  of  the  infinite  diagram  the 
method  shown  in  Fig.  13  can  be  used.  Suppose  it  is 


O       M'a' 


O        a'  L'  P' 


FIG.  13. 


desired  to  find  the  difference  of  intrinsic  energy  between 
a  and  6.  Draw  through  a  (pu-plane)  the  isodynamic 
Ea=c  and  through  6  the  iseritrope  <j>b  —  c,  interescting 
at  some  point  N.  If  the  gas  be  imagined  to  expand 
isentropically  from  6  to  N  it  will  develop  the  work 
bNN'b'  and  suffer  a  decrease  of  internal  energy  Eb—EN 
equal  to  this  work.  But  by  construction  EN=Ea  so 
that  the  area  under  bN  represents  the  magnitude  of  the 
difference  of  internal  energy  between  a  and  6.  An 


PERFECT  GASES.  35 

equivalent  solution  indicated  by  dotted  Fines  may  also 
be  used. 

In  the  T^-plane  a  slightly  different  construction 
must  be  used.  During  p.  constant  volume  change  no 
external  work  is  performed  and  the  heat  added  in- 
creases the  internal  energy.  Draw  through  b  (T1^- 
plane,  Fig.  13)  the  constant  volume  curve  Vb  =  c,  and 
through  a  the  isodynamic  Ea  =  c  intersecting  at  P. 
Tf.  the  gas  be  imagined  to  undergo  the  change  Pb  it 
will  absorb  the  heat  ct)(776-7Tp)=area  under  Pb,  and 
during  this  change  its  internal  energy  will  be  increased 
from  Ep  to  Eb,  that  is,  from  Ea  to  Eb.  Hence  the 
difference  between  the  heat  equivalents  of  the  internal 
energy  at  the  points  a  and  6  is  represented  by  the  area 
under  Pb.  An  equivalent  solution  indicated  by  dotted 
lines  may  also  be  used. 

It  follows  from  the  first  law  of  thermodynamics,  as 
applied  to  reversible  processes,  Q  =  A-AE  +  AW,  that 
(in  the  pv-plane)  since  the  area  under  db  represents  the 
work  performed  and  the  area  under  bN  represents  the 
change  of  internal  energy,  the  algebraic  sum  of  these, 
or  the  area  under  dbN,  must  represent  in  work  units 
the  heat  received.  Similarly,  in  the  TV-plane  if  the 
area  under  ab  represents  Qab,  and  the  area  under  Pb 
represents  A(Eb—Ea),  then  the  area  abPP'a'  must 
represent  AWab. 

In  the  case  of  a  perfect  gas  it  is  thus  possible  in  both 
the  pv-  and  the  T^-planes  to  represent  by  finite  areas 


36         THE   TEMPERATURE-ENTROPY  DIAGRAM. 

all  three  terms  involved  in  the  statement  of  the  first 
law  of  thermodynamics.  But  again  the  T^-plane 
proves  itself  the  simpler  of  the  two,  in  that  one  of. 
the  two  necessary  construction  curves  is  a  straight  line, 
and  the  other  has  only  one  form  for  any  given  gas. 
Graphical  Projection  of  any  Curve  from  the  pv-  to 

T>m 

the  TV- Plan 3. — From  the  laws  of  a  perfect  gas  v  =  — • 
or  vaT  if  p  =  constant,  it  follows  that  in  the  TVplane 


FIG.  14. 


a  constant  pressure  curve  is  represented  by  a  straight 
line  passing  through  the  origin.  Let  1,  Fig.  14,  repre- 
sent the  condition  T1}  v\,  and  hence  the  condition  pi. 


PERFECT  GASES.  37 

Then  the  line  01  represents  the  line  of  constant  pressure 
pi.  Now  if  the  volume  of  a  perfect  gas  be  maintained 
constant  the  pressure  is  proportional  to  the  tempera 
ture,  so  that  the  point  27\,  vi,  represents  the  pressure 
p2  =  2pi.  The  line  Op2  therefore  represents  the  con- 
stant pressure  2pi.  Similarly  Op3  represents  p  =  3pi, 
etc.  Thus  to  obtain  the  constant  pressure  curves,  lay 
off  on  any  convenient  constant  volume  curve  a  series 
of  equal  intervals  and  draw  a  set  of  straight  lines  from 
these  points  to  the  origin.  If  the  scale  of  T  is  already 
determined  the  scale  of  pressure  may  be  determined  to 
correspond,  or  if  these  pressures  are  laid  off  arbitrarily 
the  temperature  scale  must  be  determined  to  corre- 
spond. For  ordinary  use  the  latter  method  is  more 
convenient. 

Let  a  curve  ah  be  given  in  the  pv-plane  which  inter- 
sects the  pressure  curves  p\,  2pi,  3pi,  etc.,  in  the  points 
1,  2,  3,  etc.,  respectively.  Each  point  piVi,  p2V2, 
p3v3,  . . .  ,  has  an  unique  location  in  each  plane,  namely, 
at  the  intersection  of  the  corresponding  pressure  and 
volume  curves.  The  position  in  the  TVplane  may  be 
found  by  projecting  upward  along  the  constant  volume 
curve  from  p  in  one  plane  to  p  in  the  other. 

For  any  known  weight  of  gas  T  is  at  once  deter- 
mined, but  if  the  quantity  is  unknown  and  if  the 
volumes  are  only  known  relatively,  as  in  the  case  of 
cards  taken  from  hot-air  or  gas-engines  or  air-com- 
pressors, the  scale  of  T  cannot  be  determined,  but  the 


38         THE  TEMPERATURE-ENTROPY   DIAGRAM. 

projection  still  gives  relative  values  of  the  temperature 
and  may  thus  throw  considerable  light  upon  the  nature 
of  the  operations. 

The  Graphical  Projection  of  any  Curve  from  the 
TV-Plane  into  the  T^-Plane.  —  The  entropy,  temperature 
and  volume  of  a  gaseous  mixture  are  connected  by  the 
equation 

<£  =  d,  loge  T  +  AR  loge  v  +  constant. 

Assuming  any  condition  vQTQ  as  the  reference  point 
"he  difference  of  entropy  between  this  reference  point 
and  any  other  point,  vT,  is  given  by 

T  v 

4$  =  Cv  loge  TfT  +  AR  loge  -- 
1  o  VQ 

Putting  the  variable  factor  R  into  the  left-hand 
term  the  expression  reduces  to 


In  the  case  of  diatomic  gases  when  k=  1.405  this  re- 
duces to 

333  5  TV 

~^-  •  4<f>  =  2.47  logio  TfT  +  logio  -• 

ft  1  o  VQ 

This  equation  will  apply  directly  to  hot-air  engine 
and  air-compressor  cards,  but  for  gas-engine  work  the 
coefficient  may  need  to  be  modified  for  each  individual 
case,  as  A;  is  a  variable,  being  about  1.38. 


PERFECT  GASES.  39 

It  is  evident  that  if  we  could  construct  two  curves 
such  that  the  ordinates  for  any  point  (T,  v)  are  re- 

spectively equal  to  2.47  logic  TJT  and  logic  —  ,  then  the 

J-Q  ^o 

333  5 

sum   of  these  two   ordinates  would   equal   —  ^  —  J<£, 

ZV 

and  by  using  this  value  and  T  a  7^-plot  could  be  con- 
structed.   This  coefficient  could  be  eliminated  from  the 

333  5 

T<j>-plot  by  making        '   =1  the  unit  of  entropy,  and 

then   the   abscissae   would   read   directly   in   units   of 
entropy. 

Thus  suppose  we  have  the  curve  ab  (Fig.  15)  in  the 
TVplane  and  desire  to  find  its  ^-projection.  Choose 
any  point  T0v0  as  the  reference  point,  and  then  on  base 
lines  parallel  to  the  T  and  v  axes  and  located  if  necessary 
outside  the  diagram  (the  axes  themselves  may  be  used 
if  the  diagram  is  not  unnecessarily  complicated 
thereby),  construct  two  logarithmic  curves  (see  pp.  23 
to  26),  such  that 

T  v 

Ay  =  log  YQ    and     Ay  '2  =  log  —  • 

Then  by  means  of  the  auxiliary  construction  draw  also 
the  curve 


Now, 


Jjf!  =  2.47  log      - 
333.5 


40          THE  TEMPERATURE-ENTROPY  DIAGRAM. 


PERFECT  OASES.  41 

and  therefore  if  from  any  point  n  on  the  curve  ab  we 
obtain  by  projection  upon  the  two  logarithmic  curves 
the  distances  Ay^  and  Ay2,  the  sum  of  these  distances 
laid  off  to  the  right  of  (f>0  along  the  isothermal  Tn  will 
locate  the  point  in  the  T</>-plane. 
For  point  a, 

Ay  i  =  Ay  a    and     Ay2  =  Q, 

and  for  point  b, 

Ay  i  =  -  4ybl    and     Ay2  =  +  Ayb,. 

After  the   curve  is  once   constructed   the  scale   of 

333  5 
entropy  can  be  chosen  by  setting        '  •  =  !. 

It  is  evident  when  the  logarithmic  curves  are  con- 
structed graphically  that,  as  D  is  chosen  arbitrarily, 
the  scale  of  the  drawing  may  be  anything.  In  fact, 
the  scale  can  be  suitably  chosen  by  varying  the  length 
D.  Furthermore  the  values  Ayi  and  Ay2  may  be  read 
from  the  base  lines  on  which  the  logarithmic  curves 
were  constructed  or  from  any  convenient  parallel  line. 
This  would  mean  increasing  or  decreasing  all  the  values 
of  J<£  by  the  same  amount  and  result  simply  in  a  bodily 
transfer  of  the  T^-projection  to  the  right  or  left  by 
that  amount.  This  would  not  affect  the  value  of  the 
diagram,  as  we  are  dealing  with  changes  in  entropy  and 
not  with  absolute  values.  In  fact  the  value  of  <£  is 
infinite  so  that  any  point  may  be  taken  as  the  arbitrary 
zero. 


42         THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Advantage  may  thus  be  taken  of  these  facts  to 
shift  the  T ^-projection  parallel  to  the  $>-axis  so  as  to 
obtain  the  most  convenient  location  on  the  drawing- 
paper. 


CHAPTER  III. 

THE  TEMPERATURE-ENTROPY  DIAGRAM  .FOR 
SATURATED  STEAM. 

DUE  to  the  very  slight  variations  in  the  volume  of 
water  with  increasing  temperature  the  heat  equiva- 
lent of  the  external  work  is  very  small,  and  the  differ- 
ence between  the  work  performed  under  atmospheric 
pressure  and  that  which  would  be  performed  under  a 
pressure  increasing  with  the  temperature  according 
to  Regnault's  pressure-temperature  curve  is  negli- 
gible compared  with  the  heat  required  to  increase  the 
temperature.  Hence  the  value  of  the  specific  heat 
c  =  /(0  as  determined  by  Rowland  is  taken  as  equiva- 
lent to  that  of  the  actual  specific  heat  required  for  the 
transformation  occurring  when  feed-water  is  heated  in 
a  boiler.  Similar  statements  hold  for  other  fluids. 

Therefore  the  heat  required  to  increase  the  tem- 
perature of  a  pound  of  liquid  by  the  amount  dT  while 
the  pressure  increases  by  dp  is  taken  as 

dq  =  cdT,    .......       (1) 

43 


44          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Whence  q,  =      \dT,    .....      (2) 


and 


To  make  the  temperature-entropy  chart  conform 
to  the  tables  for  steam  and  other  saturated  vapors, 
we  will  plot  the  increase  of  entropy  and  intrinsic 
energy  and  the  heat  added  above  that  of  the  liquid 
at  freezing-point.  Hence  for  water  the  zero  of  the 
entropy  scale  will  be  at  32°  F.  Furthermore  the  chart 
will  be  constructed  for  one  pound  of  water. 

Point  a  in  Fig.  16  represents  the  position  of  a  pound 
of  water  at  32°  F.  on  the  !T0-chart.  Starting  from 
here  and  plotting  values  of  6  and  T  from  the  equation 


6- 

32° 


as  given  in  the  steam- tables,  we  obtain  the  "water- 
line  "  ab.  At  any  temperature  t  the  value  of  the  specific 
heat  c  is  shown  by  the  subtangent  gf.  Further,  as  we 
proved  in  the  general  case  of  equatipn  (1),  the  area 
under  the  curve  aefO  represents  the  heat  of  the  liquid, 
q;  that  is,  the  number  of  heat-units  required  to  warm  up 
the  water  from  32°  F.  to  the  temperature  t.  This 
"water-line"  is  the  same  kind  of  a  curve  as  that  repre- 
sented by  equation  (15)  for  perfect  gases,  except  that 


SATURATED  STEAM. 


45 


the  specific  heat  of  a  gas  is  a  constant,  while  that  of 
water  is  a  function  of  the  temperature. 

If  at  t  the  water  has  reached  the  temperature  corre- 
sponding to  the  boiler  pressure  any  further  increase 
of  heat  will  cause  the  water  to  vaporize  under  con- 
stant pressure  and  thus  there  will  be  an  increase  of 
entropy  at  constant  temperature.  This  will  be  repre- 
sented on  the  chart  by  the  horizontal  line  ed,  and 
will  continue  until  all  the  water  is  vaporized  and  a 


82' 


-f 


9    O      f  m    $. 

FIG.  16. 

condition   of   dry-saturated   steam   reached.      During 
this  change  the  increase  of  entropy  will  be 


that  is,  the  length  of  the  line  ed  may  be  found  by  taking 
the  latent  heat  of  vaporization  r  and  dividing  by  the 
absolute  temperature  corresponding  to  t. 


46  THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  area  under  the  curve  ed  represents  the  heat 
added  during  vaporization  or  r. 

The  area  under  the  curve  aed  therefore  represents 
all  the  heat  necessary  to  be  added  to  a  pound  of  water 
at  32°  F.  to  change  it  in  a  boiler  into  dry  steam  at 
the  temperature  t,  or  area  Oaedm  =  X  =  q+r. 

Similarly  the  location  of  the  "dry  -steam"  point  may 
be  found  for  any  number  of  temperatures,  thus  giving 
the  location  of  the  dry-steam  line  or  saturation  curve. 

In  the  area  between  the  water-line  and  the  dry- 
steam  line  ij,  or  the  region  of  vapor  in  contact  with 
its  liquid,  lie  all  the  values  given  in  the  steam-tables;  to 
the  right  of  the  dry-steam  curve  lies  the  region  of 
superheated  steam. 

Having  given  the  chart  with  the  "  water-line"  and 
"dry-steam"  line  located,  and  knowing  the  temperature 
t  of  the  steam,  it  is  simply  necessary  to  draw  the  hori- 
zontal line  ed  and  drop  the  two  perpendiculars  ef  and 
dm  and  the  tangent  eg-,  then  the  diagram  gives  at  once 


q,  r,  ;,  0,     >  T,  and  c. 


Let  e  in  Fig.  17  represent  the  state  point  of  a  pound 
of  water  in  a  boiler  under  the  pressure  p  corresponding  to 
the  temperature  t.  As  heat  is  applied  part  of  the 
water  vaporizes  and  the  state  point  moves  toward  d. 
At  any  point  as  M,  the  area  under  eM  represents  the 
heat  already  added,  while  the  remaining  part,  under  Md, 


SATURATED  STEAM. 


47 


represents  the  heat,  which  must  be  added  to  complete 
the  vaporization.  That  is,  for  the  complete  change  from 
water  to  dry  steam  the  state  point  travels  the  distance 
ed,  and  to  vaporize  one  half  it  would  travel  one  half 
the  distance,  etc.  Then  if  M  represents  the  momentary 
position  of  the  state  point,  the  ratio  eM  +  ed  will  repre- 
sent the  fractional  part  of  the  water  already  vaporized; 


O  N 

FIG.  17. 


that  is,  the  dryness  of  the  mixture,  x,  is  given  by 
eM 


ed 


(6) 


Hence  if  the  line  ed  is  divided  into  any  convenient 
number  of  equal  parts  (say  10)  the  value  x  can  be  read 
directly  from  the  chart  as  soon  as  the  position  of  the 
state  point  M  is  known.  In  Fig.  17,  for  example, 


48          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Furthermore,  the  value  of  the  entropy  at  M  is  repre- 
sented by  that  of  the  liquid  at  e  and  the  xih  part  of 
the  entropy  of  vaporization,  or 


and  the  total  heat  at  M  equals  the  heat  of  the  liquid 
plus  the  xih  part  of  the  heat  of  vaporization,  or 


and  is  represented  by  the  area  OaeMN. 

In  a  similar  way  the  distance  -^  for  several  tem- 

peratures can  be  divided  into  the  same  number  of 
equal  parts,  and  then  if  all  these  corresponding  points 
are  connected  by  smooth  curves,  each  curve  will  repre- 
sent a  change  during  which  the  fractional  part  of  the 
water  vaporized  is  constant.  These  are  known  as  the 
z-curves. 

In  place  of  having  separate  scales  of  pressure  and 
temperature  for  the  ordinates  of  the  !F<£-diagram, 
it  is  often  convenient  to  take  the  values  of  p  and  t  from 
the  steam-tables  and  to  plot  Regnault's  pressure- 
temperature  curve  in  the  second  quadrant,  as  shown  in 
Fig.  18. 

Then  given  any  pressure  p^  the  corresponding  tem- 
perature ij  may  be  found  as  indicated  or  vice  versa. 


SATURATED  STEAM. 


49 


Let  Pi  be  the  pressure  and  xt  the  dryness  fraction 
of  a  pound  of  mixture,  then  on  the  chart  its  condi- 


Fia  18. 


tion  will  be  indicated  by  the  point  1.     Its  volume  may 
be  found  from  the  steam-tables  by  use  of  the  formula 


50          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

To  find  the  location  of  the  state  point  3  at  any  other 
pressure  p3,  so  that  v3  =  i\,  proceed  as  follows: 

Draw  the  axis  of  volume  opposite  to  that  of  tem- 
perature and  lay  off  along  Ov  the  distance  Oa  equal 
to  the  volume  of  one  pound  of  water.  The  variations 
of  a  with  t  may  be  neglected  and  o  set  equal  to  0.016 

cubic  feet.    Draw  oa.    Project  6V  and  6^+-^-  on  the 

^-axis,  and  from  the  latter  point  lay  off  the  distance 
msx  equal  to  the  volume  of  one  pound  of  saturated 
steam  as  given  in  the  steam-table  for  t°.  Prolong  the 
perpendicular  from  0t  until  it  intersects  aa  at  ov.  Con- 
nect this  last  point  with  sv  This  line  alsl  shows  the 
increase  of  the  volume  and  the  entropy  during  vapori- 
zation. Project  xl  upon  0<j>  and  continue  until  it 
intersects  the  rx^-curve  at  c.  Then  be  =  x^  and  ac  =  x^ 
+  o1  =  v1.  Through  c  draw  yy  parallel  to  0$.  For  any 
pressure  p3  draw  the  corresponding  t'0-curve  <73.?3,  and 
where  this  intersects  yy  the  volume  will  be  V3'=v1=x3u3 
+  o.  Projected  up  this  intersects  the  isothermal  t3 
in  3,  giving  the  desired  dryness  fraction  x3.  Points  1 
and  3  have  the  same  volume  i\.  Other  points,  as 
2,  etc.,  may  be  found  and  connected  with  a  smooth 
curve.  This  will  intersect  the  dry-steam  line  at  some 
point  s9  =  vl.  In  this  manner  similar  constant  vol- 
ume curves  can  be  constructed  to  cover  the  entire 
diagram. 
Suppose  it  is  required  to  find  the  heat  necessary 


SATURATED  STEAM.  51 

to  cause  a  change  from  some  point  L  to  an  adjacent 
point  L  +  dL.     (See  Fig.  19.) 

,     a  ,  xr      rcdT    xr 
v  =  °+Y=J  ~T~  +  ~T' 

cdT     x  ,       r          xr 


OCT 

-dT  +  (cxdT  -  cxdT) 


(7) 


To  interpret  the  last  term  imagine  this  change  to 
occur  along  the  dry-steam  line.  Then  x  =  l  and 
dx  =  Q,  whence 


dQ=\c—Y+£p]dT(  =  Td<f>).      .    .     (7fl) 

The  comparison  of  this  with  eq.  (2),  p.  4,  shows  that 
c—Tr+Tm  'ls  tne  "specific  heat"  of  dry-saturated  steam, 

r      dr 
or  h^c—-Ffi  +  -rFn' (8^ 


The  diagram  shows  at  once  that  h  is  negative,  i.e. 
to  move  along  the  dry-steam  line  with  increasing  tem- 
perature heat  must  be  rejected. 

Ether  has  a  positive  value  for  h.  This  signifies  that 
the  saturated-vapor  line  for  ether  slants  to  the  right 
instead  of  to  the  left. 


52 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


Him  found  that  steam  condensed  upon  adiabatic 
expansion  and  Cazin  that  it  did  not  condense  upon 
compression.  The  reason  for  this  is  at  once  clear  from 
Fig.  19.  Expanding  from  a  the  reversible  adiabatic 


Water 


FIG.  19. 


line  for  water  cuts  successively  ' '  z-lines  "  of  decreasing 
value,  showing  condensation.  Compressed  adiabatic- 
ally  from  a  the  steam  would  at  once  become  super- 
heated. Exactly  the  reverse  occurs  with  ether,  con- 
densation occurring  during  adiabatic  compression,  super- 
heating during  expansion. 

If  water  is  permitted  to  expand  adiabatically  from 
6  it  is  partially  vaporized,  as  shown  by  the  adiabatic 
line  cutting  increasing  values  of  x.  Similarly  if  very 
wet  steam  is  compressed  it  condenses. 


SATURATED  STEAM.  53 

An  inspection  of  the  z-curves  near  the  value  z  =  0.5 
shows  that  they  change  from  convex  to  concave  and 
that  it  is  thus  possible  with  water  for  the  reversible 
adiabatic  to  cut  an  x-curve  twice  at  different  tempera- 
tures, as  at  c  and  d;  i.e.,  it  is  possible  at  the  end  of  an 
isentropic  expansion  to  have  the  same  value  of  x  as  at 
the  beginning.  Thus 

t,  =  4'1=Ol+=0,+Z£i,    or    *,--.      0) 


Consequently  there  must  exist  some  adiabatic  which 
is  tangent  to  this  z-curve.  Above  the  point  of  tan- 
gency  the  z-curve  slants  to  the  right  and  possesses  a 
positive  specific  heat,  below  it  the  curve  slants  to  the 
left  and  has  a  negative  specific  heat.  The  values  of 
the  specific  heat  above  and  below  the  point  of  tan- 
gency  diminish  in  magnitude  as  the  tangent  is  ap- 
proached and  at  the  point  of  tangency  are  identical 
and  equal  to  zero;  that  is,  just  there  the  tempera- 
ture may  be  raised  without  the  addition  of  heat  because 
the  change  is  isentropic. 

If  these  points  of  zero  specific  heat  are  determined 
for  all  the  x-curves  and  connected  by  the  smooth  curve 
MN,  the  chart  is  divided  into  two  portions,  such  that 
to  the  left  of  MN  the  specific  heats  along  the  z-lines  are 
positive,  i.e.  isentropic  expansion  will  be  accompanied 
by  vaporization,  and  to  the  right  of  MN  the  specific 


54  THE  TEMPERATURE-ENTROPY  DIAGRAM. 

heats  are  negative  and  isentropic  expansion  will  be 
accompanied  by  condensation.  The  curve  MN  is 
known  as  the  zero-curve. 

So  far  we  have  located  p,  v,  T,  <£,  and  x,  and  to  make 
the  definition  of  the  point  complete  it  is  only  necessary 
to  draw  a  curve  of  constant  intrinsic  energy.  This 
could  be  done  by  solving  E1  =  E2==E3  =  etc.,  =  ql  +X1p1=q2 
+X2p2  =  q3+x3p3  =  etc.,  for  the  proper  values  of  x  and 
connecting  these  by  a  smooth  curve.  This  would  be 
very  laborious,  as  there  does  not  seem  to  be  any  con- 
venient graphical  construction.  Fortunately  it  is  pos- 
sible not  to  draw  the  isodynamic  curves,  but  to  find 
an  area  which  represents  the  value  of  the  intrinsic 
energy  for  any  state  point  and  at  the  same  time  to 
divide  r  into  two  areas  proportional  to  p  and  Apu 
respectively. 

The  first  fundamental  heat  equation 


becomes,  when  made  to  conform  to  the  limitations  of 
the  first  and  second  laws  of  thermodynamics, 


This  is  applicable  to  the  case  of  saturated  vapors 
because  the  state  point  is  uniquely  defined  by  the 
intersection  of  the  temperature  and  volume  curves. 


SATURATED  STEAM.  55 

Equating  the  coefficient  of  the  last  terms, 


a  relationship  which  holds  good  for  any  reversible 
change.  During  the  process  of  evaporation  T  ic  a 
constant  and 

«)  =_^  or  r_A7&,        (10) 

/T     s  —  a  u  dT 


whence  Apu-^-  ....     (10a) 

dT7        pdT 

Let  a  (Fig.  20)  be  the  point  of  which  the  intrinsic 
energy  is  to  be  obtained,  pa  is  the  corresponding  pres- 
sure. From  of  on  Regnault's  !Tp-curve  draw  the  tan- 
gent a'b'.  From  similar  triangles  it  is  evident  that 

jrp 

the  subtangent  equals  p  •  -T-.     Draw  b'b  parallel  to  a'a. 

jm 

Then  the  rectangle  abed  has  the  area  p--j-X-^  =  Apu 

(see  eq.  10a). 

That  is,  abed  represents  the  external  heat  of  vapori- 
zation and  the  rest  of  r,  namely  bckf,  must  equal  p. 
Hence  the  intrinsic  energy  of  the  point  a  is  given  by 

Ea  =  OgdW  +  kcbf  =  q  +p. 


56          THE  TEMPERATURE-ENTROPY  DIAGRAM. 
If  M  represents  the  state  point 
dMnc  =  xMApu, 

and 

If  for  each  point  a  of  the  dry-steam  line  a  correspond- 
ing point  c,  which  divides  the  absolute  temperature 


FIG.  20. 

into  two  parts  proportional  to^Apu  and  p,  is  determined 
the  curve  ee  will  be  located.  The  curve  ee  being  located 
once  for  all,  the  intrinsic  energy  at  any  point  M  can  be 
found  by  drawing  the  isothermal  Md  and  the  adiabatics 
dk  and  Mm.  At  c,  the  point  of  intersection  of  dk  with 
ee,  draw  the  isothermal  en.  Area  OgdcnmO  gives  the 
desired  value. 


SATURATED  STEAM. 


57 


The   temperature-entropy   diagram   for   steam   thus 
enables  one  to  find  directly  the  following  quantities: 

p,  t,  v,  E,  <j>,  x,  c,  h,  s,  q,  r,  I,  p,  6,  ^,  and  Apu.    That 

is,  the  diagram  is  equivalent  to  a  set  of  steam-tables 
and  in  some  ways  superior  to  them  in  that  it  enables 
one  to  obtain  a  comprehensive  idea  of  the  changes 
taking  place  between  these  quantities. 
Let  the  curve  MN  in  Fig.  21  represent  the  T^-pro- 


FIG.  21. 

jection  of  some  reversible  change.  During  the  change 
MN  there  has  been  added  from  some  external  source 
the  amount  of  heat  MNnm.  At  the  same  time  the 
intrinsic  energy  has  increased  from  OgabcmO  to 
OgdefnO,  the  increase  being  shown  by  the  area 
adefnmcba.  The  area  MifnmM  is  common  to  both  the 
heat  added  and  the  increase  of  intrinsic  energy,  and 
as  the  remaining  part  of  the  intrinsic  energy  increase, 
adeiMcba,  is  greater  than  the  remaining  portion  of  the 
heat  added,  iNfi,  it  follows  that  the  heat  added  does 


58 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


not  equal  the  increase  of  internal  energy  and  hence 
an  amount  of  work  must  have  been  performed  upon  the 
substance  equal  to  the  difference  in  area  of  these  two 
surfaces;  i.e.,  the  external  work  performed  upon  the 
substance  equals  adeiMcba—iMfi.  These  areas  may  be 
found  readily  with  a  planimeter. 

The  performance  of  external  work  upon  the  sub- 
stance might  have  been  foretold  at  once  from  the 
diagram,  because  the  volume  of  N  is  less  and  the  pres- 
sure greater  than  that  for  M. 

Fig.  22  represents  another  type  of  reversible  change. 
During  the  transformation  MN,  there  is  added  the 


FIG  22. 

heat  MNnm,  and  the  intrinsic  energy  changes  from 
OgabcmO  at  M  to  OgdefnO  at  N.  The  portion 
OgdeimO  is  common  to  both,  and  the  intrinsic  energy 
at  N  will  be  greater  or  less  than  it  was  at  M  according 
as  the  area  ifnmi  is  greater  or  less  than  the  area  abcieda. 
The  magnitude  and  sign  of  this  difference  may  be  de- 
termined each  time  by  the  use  of  planimeters. 


SATURATED  STEAM.  59 

In  Fig.  22,  En>En]  hence  this  increase  of  internal 
energy  must  have  come  from  the  heat  added,  and 
subtracting  this  difference  from  the  total  heat  added 
will  give  the  amount  remaining  for  external  work. 
This  is  positive  in  the  case  shown,  as  was  to  be  expected, 
as  the  pressure  has  fallen  and  the  volume  increased. 

In  case  there  was  a  decrease  in  the  internal  energy 
that  area  would  need  to  be  added  to  the  heat  area  in 
order  to  obtain  the  external  work  performed. 

Having  learned  to  construct  and  interpret  the  T$- 
diagram  for  saturated  vapors  we  must  now  resume 
once  more  the  main  object  of  our  investigation,  namely, 
to  find  the  location  in  the  T^-plane  of  any  curve  given 
in  the  pv-plaue  or  vice  versa,  so  that  we  may  eventually 
be  able  to  investigate  the  action  of  a  steam-engine  from 
both  its  indicator  and  its  temperature-entropy  diagrams. 

In  the  case  of  perfect  gases  it  was  possible  to  use 
one  of  the  fundamental  heat  equations  and  thus  obtain 
a  simple  analytical  expression  which  could  be  easily 
*  plotted;  for  saturated  steam,  however,  this  process  is. too 
cumbersome  to  be  of  any  service.  Fortunately  the 
graphical  method  offers  a  solution  both  simple  and 
elegant. 

In  constructing  the  T</>-diagram  we  have  already 
made  use  of  the  first,  second,  and  fourth  quadrants  to 
express  T<f>,  pt,  and  v<f>  variations  respectively,  and  now 
we  have  but  to  take  the  third  or  pv-quadrant  and  the 
diagram  is  complete. 


60          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  saturation  curve,  or  curve  of  constant  steam- 
weight  ab  in  the  pv-plane,  is  depicted  by  the  dry-steam 
curve  a'br  in  the  jT</>-plane,  Fig.  23.  The  method  of 
obtaining  corresponding  points  aaf  and  W  is  shown 
by  the  projection  through  the  point  al  and  b1  of  the 
pT-curve.  Aa  and  Bb  show  the  increase  in  volume  of  a 
pound  of  H20  m  vaporizing  under  the  constant  pres- 
sure pa.  and  pi>  respectively.  This  same  increase  of  vol- 
ume is  represented  in  the  <£r-plane  by  the  curves  A" a" 
and  B"V  respectively.  Now  if  only  part  of  the  pound 
is  vaporized  the  actual  volume  will  be  indicated  by 
the  points,  say,  xa  and  xb.  By  projection  into  the 
emplane  xa"  and  xb"  are  found.  Another  projection 
and  we  obtain  xar  and  Xb  as  the  state  points  in  the 
T^-plane  corresponding  to  the  points  xa  and  xb  in 
the  pv-plane. 

Suppose  the  pressure  and  volume  of  a  pound  of 
steam  have  been  determined  for  some  particular  part 
of  the  stroke  by  means  of  the  indicator-card  and  steam 
measurements.  Let  xa  represent  such  a  point.-  It  is 
now  desired  to  find  the  corresponding  state  point  in 
the  !T0-plane.  The  procedure  is  as  follows: 

Draw  through  xa  the  curve  of  constant  pressure  Aa 
and  determine  by  projection  its  location  A  'a'  and 
A"a"  in  the  T$-  and  0v-planes  respectively.  Then 
project  xa  to  xa"  and  finally  xa"  to  xa'  and  the  desired 
determination  is  finished. 

As  a  further  problem  suppose  it  is  desired  to  locate 


SATURATED  STEAM. 


61 


on  the  T<£-plane  the  curve  of  constant  volume  xaxc. 
The  point  xa  is  already  located  at  xd .  To  locate 
xc  draw  Bb  and  find  its  projections  at  B'b'  and  B"b". 


FIG.  23. 


Then  project  xc  to  x"  and  finally  to  xef.  The  two 
end-points  of  the  curve  being  determined,  any  inter- 
mediate point  as  xd  will  be  located  in  the  same  manner 
as  shown.  Thus  after  a  sufficient  number  of  points 


62          THE  TEMPERATURE-ENTROPY  DIAGRAM. 

are  located,  the  curve  of  constant  volume  xa'xc'  may 
be  drawn.  Naturally  if  the  chart  is  already  provided 
with  constant-volume  curves  this  construction  would 
be  unnecessary. 

Passing  now  to  the  most  general  problem  consider 
the  curve  LMN  and  suppose  its  equation  in  the  pv- 
plane  to  be  pvn=plvln.  This  would  correspond  to  the 
expansion-line  of  an  indicator-card.  It  is  desired  to 
find  the  projection  of  this  curve  in  the  !T</>-plane. 
The  problem  resolves  simply  into  the  location  of  a 
sufficient  number  of  state  points,  through  which  a 
smooth  curve  is  finally  to  be  drawn.  To  locate  L,  M, 
and-  N  project  them  on  to  the  corresponding  volume 
curves  A" a",  7)"d",  and  B"V  of  the  <£v-plane  at  L", 
M",  and  N" ,  and  then  finally  project  to  L',  M',  and  Nf. 
To  properly  determine  the  curve  some  intermediate 
point  as  K  may  be  necessary. 

The  General  Method  of  Representing  AE  and  W. — 
The  method  just  explained  is  simple  in  application 
but  is  confined  to  saturated  vapors  alone.  It  is,  how- 
ever, possible  by  a  line  of  reasoning  similar  to  that 
adopted  for  perfect  gases,  to  develop  a  method  of  repre- 
senting AE  and  W  which  may  be  applied  to  saturated 
and  superheated  vapors  alike. 

Suppose  it  is  desired  to  find  EA,  xApuA,  and  H  for 
a  pound  of  saturated  vapor  whose  state  point  is  at  A 
(Fig.  24).  Through  g  and  A  draw  the  isodynamic 
curves  Eg=Q  and  EA  =  c,  respectively,  and  also  draw 


SATURATED  STEAM. 


63 


the  curve  of  constant  volume  VA  =  c  intersecting  Eg  =  Q 
in  the  point  C.  During  the  constant  volume  change 
no  external  work  is  performed  so  that  the  area  ncAmn 
represents  the  increase  of  internal  energy  in  going  from 
g  to  A,  and  hence  is  equal  to  qa  +  xpA  =  EA. 

It  is  possible  to  divide  this  area  into  two  parts  repre- 
senting qa  and  xpA,  respectively,  by  finding  the  inter- 
section B  of  the  isodynamic  Ea  =  qa,  drawn  through  a, 


"n        i  m  <t> 

FIG.  24. 

with  the  constant  volume  curve  AC.  Draw  the  per- 
pendicular El.  Then  area  cBln  represents  the  heat  of 
the  liquid  qa,  and  area  BAml  represents  the  intrinsic 
energy  of  vaporization  xp. 

Since  the  area  OgaAmO  represents  the  total  heat  HA 
equal  to  qa  +  xpA  +  xApuA  it  follows  that  the  difference 
between  these  two  areas,  or  the  area  OgaACnO.  must 
represent  the  external  heat  of  vaporization. 

Fig.  24  illustrated  the  special  problem  of  represent- 
ing the  quantities  of  the  steam  tables  by  assuming  the 


64 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


substance  to  start  from  freezing  and  to  reach  the  state 
point  A  by  travelling  along  the  curve  gaA,  and  is  readily 
seen  to  represent  a  special  case  of  the  more  general 
problem  illustrated  in  Fig.  25. 


Let  it  be  required  to  find  QAB,  AWAB,  and  4E 'AB 
for  any  general  process  such  as  AB.  Draw  through  A 
the  isodynamic  curve  AC  and  through  B  the  constant 
volume  curve  BC  intersecting  at  C.  Then  if  the  vapor 
be  assumed  to  undergo  the  process  CB,  the  increase  in 
intrinsic  energy  EB—EA  will  be  represented  by  area 
CBbc.  And  as  QAB  is  represented  by  area  ABba,  the 
heat  equivalent  of  the  external  work  must  be  repre- 
sented by  area  ABCca. 

It  is  true  that  the  isodynamic  through  g  cannot  be 
drawn  without  knowledge  of  the  properties  of  water 
vapor  in  contact  with  ice,  but  this  does  not  affect  the 
validity  of  this  method  as  applied  to  the  general  case 
just  discussed. 


CHAPTER  IV. 

THE  TEMPERATURE    ENTROPY  DIAGRAM  FOR  SUPER. 
HEATED  VAPORS. 

THE  analytical  treatment  of  superheated  vapors,  such 
as  steam,  ammonia,  sulphurous  anhydride,  etc.,  is 
more  complicated  than  that  of  perfect  gases  for  two 
reasons  : 

(1)  The    characteristic    equations    are    more    com- 
plicated and  but  imperfectly  known;  and 

(2)  The  specific  heat  at  constant  pressure  and  con- 
stant volume   are  no  longer  constant  but  follow  com- 
plicated laws,  unknown  in  most  cases  and  but  imper- 
fectly known  in  others. 

At  the  present  time  two  different  equations  are  being 
used  for  superheated  steam.  Thus  Stodola  in  his 
Steam  Turbines  makes  use  of  the  Battelli-Tumlirz 
equation  p(v  +  Q.135)=85.1T,  while  Peabody  has  based 
his  new  Entropy  Tables  upon  the  equation  of  Knob- 
lauch, Linde.,  and  Klebe, 


pv  =  85.8571  -  p(l  +  0.00000976?) 


A  simplified  form  of  the  latter,  which  does  riot  differ 

from  this  by  more  than  0.8  per  cent.,  has  the  same 

65 


66 


THE  TEMPERATURE-ENTROPY    DIAGRAM. 


form  as  the  Battelli-Tumlirz,  but  different  values  for 
the  constants,  viz., 

2KV  + 0.256)  =  85.857. 

Of  the  manv  determinations  of  cp  the  results  of  only 
two  sets  of  investigators  nee:l  be  noticed.  Those 
obtained  by  Thomas  and  Short,  because  their  values 
are  used  in  Peabody  's  Entropy  Tables,*  and  those  of 
Knoblauch  and  Jakob,  as  their  values  vary  most  closely 
in  accordance  with  theoretical  predictions: 

THOMAS  AND  SHORT.     (Mean  value  of  cp.) 


Decree  of 
Superheat 
Fahr. 

Pressure  Pounds  per  Square  Inch.     (Absolute.) 

6 

15 

30 

50 

100 

200 

400 

20° 
50° 
100° 
150° 
200° 
250° 
300° 

0.536 
0.522 
0.503 
0.486 
0.471 
0.456 
0.442 

0.547 
0.532 
0.512 
0.496 
0.480 
0.466 
0.453 

0.558 
0.542 
0.524 
0.508 
0.424 
0.481 
0.468 

0.571 
0.555 
0.537 
0.522 
0.509 
0.496 
0.484 

0.593 
0.575 
0.557 
0.544 
0  533 
0.522 
0.511 

0.621 
0.600 
0.581 
0.567 
0.556 
0.546 
0.537 

0.649 
0.621 
0.599 
0.585 
0.574 
0.564 
0.554 

KNOBLAUCH  AND  JAKOB.     (Mean  value  of  cp.) 


Kg. 

Si. 

sq. 
Cor.t 
Cen 
Cor. 
Fab 

F. 

212° 
302° 
392° 
482° 
572° 
662° 
752° 

per 
m... 
per 
in.... 
emp. 
t  
temp 
r.  .  . 

C. 

100° 
150° 
200° 

LT,Ori 

300° 
350° 
400° 

14.2 
99° 
210° 

0.463 

0.462 
0.462 
0.463 
0.464 
0.468 
0.473 

2 
28.4 
120° 
248° 

4 
56.9 
143° 
289° 

6 
85.3 
158° 
316° 

8 
113.8 
169° 
336° 

10 
142.2 
179° 
350° 

12 
170.6 

:ar 

368° 

14 
199.1 
194° 
381° 

16 
227.5 
200° 
392° 

18 
156.0 
206° 
403° 

20 
284.4 
211° 
412° 

0.478 
0.475 
0.474 
0.475 
0.477 
0.481 

0.515 

0.502 
0.41)5 
I).  11)2 
0.492 
0.494 

0.530 

0.514 
0.505 
0  .  503 
0.504 

0.560 

0>>32 
0.517 
0.512 
0.512 

0.597 

0.552 

0.530 

0.522 
0.520 

0.635 

0.570 
0.541 
0.520 
0.526 

0.677 

0.5SS 

0.550 
o.r,:u> 
0.531 

0.609 
0.561 

0.513 
0.537 

0.635 

0.572 
0.550 
0.542 

0.664 

0.5S5 
)  .  557 
0.547 

*  In  the  1909  and 
uses  the  values  of  cr 


later  editions  of  the  Entropy  Tables  Peabody 
obtained  by  Knoblauch  and  Jacob. 


SUPERHEATED    VAPORS.  67 

To  obtain  the  heat  required  to  superheat  at  constant 
pressure  we  must  either  know  how  cp  varies  with  the 
temperature  at  any  given  pressure  and  then  integrate 

/*2 

the  expression  Q=  I    CpdT,  or  else  we  may  do  what 

is  accurate  enough  for  all  engineering  work  —  take  a 
mean  value  of  cp  and  assume  this  value  to  be  main- 
tained throughout  the  given  temperature  range. 

Making  the  same  assumption  with  reference  to  Cp 
the  increase  in  entropy  during  superheat  is  given  by 

dT  T2 


In  superheated  as  in  saturated  steam  increase  in 
entropy  and  internal  energy  as  well  as  total  heat  added 
along  a  constant  pressure  curve,  are  all  measured  from 
water  at  32°  F.  as  an  arbitrary  zero.  Thus  if  a  pound 
of  water  at  32°  F.  is  confined  under  a  pressure  of  p 
pounds  per  square  foot,  and  is  then  heated  to  a  tem- 
perature T  corresponding  to  this  pressure,  next  vaporized 
at  constant  pressure,  and  finally  superheated  at  constant 
pressure  to  some  final  temperature  T8,  we  may  express 
the  changes  in  its  entropy,  and  internal  energy  and  the 
total  heat  added  by  the  formulae, 

TcdT    xr  T8 

2-f-  +  Y+Cplo^eT' 

=  H=q+xr+cp(Ts-T) 


68         THE    TEMPERATURE-ENTROPY  DIAGRAM. 

The  first  term  of  each  expression  refers  to  the  warm- 
ing of  the  water,  the  second  to  its  vaporization,  and 
the  rest  to  the  process  of  superheating.  Of  course, 
when  the  steam  is  superheated  x  =  l. 

There  is  no  method  of  measuring  directly  the  increase 
in  internal  energy  during  vaporization  or  during  super- 
heating at  constant  pressure,  so  that  in  both  cases  we 
are  forced  to  fall  back  upon  the  first  law  of  thermo- 
dynamics Q  =  A(AE  +  W),  and  measure  Q  and  AW, 
thus  obtaining  AAE  indirectly.  Thus  p  =  r— Apu,  and 
similarly  the  increase  of  internal  energy  from  a  condition 
of  dry  steam  to  any  degree  of  superheat  equals  Q  —  AW 
=  cp(Ta-T)-Ap(v-s). 

The  above  equations  combined  with  those  for  deter- 
mining the  specific  volume, 

r  =  .016  +  z(s-.016)  for  saturated  steam 
and      pv  =  85.85T— 0.256/>  for  superheated  steam, 

make  it  possible  to  solve  all  heat  and  work  problems 
involving  changes  of  condition  in  superheated  steam, 
or  between  saturated  and  superheated  steam. 

Construction  of  the  Constant  Pressure  Curves  in  the 
TV-Plane.— Starting  at  the  dry-steam  line  (Fig.  26)  at 
any  pressure  p  corresponding  to  the  temperature  T, 
look  up  in  the  cp-table  the  mean  value  of  cp  for  this 
pressure,  and,  say,  50  degrees  superheat.  Compute  the 
value  of 


SUPERHEATED  VAPORS. 


69 


Take  from  the  table  the  mean  value  of  Cp  for  each 
successive  50  degrees  and  compute 


T  +  150 


Continue  this  operation  until  a  sufficient  number  of 
points  has  been  determined  to  locate  the  curve  with 


FIG.  26. 


the  desired  accuracy.  This  operation  must  be  repeated 
until  a  sufficient  number  of  lines  has  been  accurately 
determined  to  permit  of  interpolation  for  the  remainder. 

If  a  copy  of  the  entropy  tables  is  at  hand  it  may  be 
used  to  plot  those  curves  or  portions  of  curves  which 
fall  within  its  limits. 

In  general  these  curves  are  about  twice  as  steep  as 
the  water-line,  as  the  specific  heat  of  water  is  about 
twice  that  of  superheated  steam. 


70         THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Construction  of  the  Constant  Volume  Curves. — It  is 
possible  from  the  fundamental  relation 

AT 


cv-cv= 


idT\  idT\  > 
\dpL\dv /p 


together  with  the  characteristic  equation  for  super- 
heated steam  and  the  tables  of  cp,  to  derive  (1)  an  ex- 
pression for  the  momentary  value  of  cv  in  terms  of 
the  momentary  values  of  p,  v,  T,  and  cp;  or  (2}  an 
expression  for  the  mean  value  of  cv  at  any  volume 
for  a  given  increase  in  tempearture.  By  such  a  method 
a  table  could  be  constructed  for  cv. 

Until  a  table  of  values  of  cv  has  been  computed  the 
simplest  method  of  constructing  the  constant  volume 
curves  is  as  follows : 

Substitute  the  value  of  the  desired  volume  in  the 
characteristic  equation  and  then  solve  the  equation 
to  obtain  the  pressures  corresponding  to  a  series  of 
temperatures  Ts,  !FS  +  50,  !TS4-100,  .  .  .  From  the 
pressures  thus  obtained  look  up  the  corresponding 
temperatures  of  saturated  steam  and  the  mean  values 
of  cp  between  the  dry-steam  line  and  Ts,  !FS  +  50,  ... 
From  these  data  compute, 

Ts  .       Ts 

J<pl  =  CPl  10ge  yT,       J02  =  Cp,  10ge  

*  93 


SUPERHEATED   VAPORS. 


71 


Plot  the  temperature  against  the  corresponding  in- 
crease in  entropy  along  the  constant  pressure  curve  from 
the  dry-steam  line  up  to  the  given  point  for  each  tem- 
perature (Fig.  27).  The  constant  volume  curve  thus 


FIG.  27. 


determined  will  of  course  be  steeper  than  the  constant 
pressure  curves  because  cv  <  cp  and  therefore 


The  work  involved  can  be  much  reduced  by  using 
the  entropy  tables.    Look  under  each  value  of  entropy 


72        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

for  the  desired  volume  and  read  at  once  the  degrees 
superheat.  Beyond  the  limits  of  the  tables  the  above 
method  must  be  used. 

Isodynamic  Curves. — The  isodynamic  line  for  a 
perfect  gas  coincides  with  the  isothermal;  for  a  mixture 
of  a  liquid  and  its  vapor  the  divergence  is  very  great, 
the  temperature  along  the  isodynamic  dropping  rapidly 
as  the  entropy  increases;  for  superheated  vapors  which 
occupy  an  intermediate  position  the  isodynamic  ap- 
proaches more  and  more  closely  to  the  isothermal  the 
further  the  state  point  is  from  the  dry-vapor  line. 

To  plot  the  isodynamic  E  =  c  (Fig.  28)  extending 


FIG.  28. 

from  the  saturated  into  the  superheated  region  we 
have 


In  the  saturated  region  the  quality 


SUPERHEATED  VAPORS.  73 

may  be  obtained  for  a  sufficient  number  of  tempera- 
tures by  looking  up  the  corresponding  q  and  p. 

In  the  superheated  region  it  is  necessary  to  find  the 
points  of  intersection  of  the  isodynamic  with  several 
constant  pressure  curves.  The  process  is  cumbersome, 
as  the  temperature  can  only  be  found  by  approxima- 
tion. Thus  given  E  =  c  and  p  =  c,  we  have 

E  =  q+P  +  cp(Ts-T)-Ap(v-s), 

where  everything  is  known  except  Ts,  v,  and  cp.     But 
from  the  characteristic  equation 


p 

By  combination  we  have 

E=q+p+cP(Ts-T)-Ap(— 0.256-sj; 

whence 

/       SS.SSX.,  (s+0.256)p 

I  Cp __„  '  )  JL  ~—  CpJ.    =  El  ~~~Q  —  P~~~  '  7 TO  ' 

\  7lO  /  '  ' O 

=AT. 

Assume  Tg  and  take  the  corresponding  mean  value 
of  cp  from  the  table.  If  the  left  hand  gives  a  value 
differing  from  N  try  other  values  of  Ts. 

Having  found  the  temperatures  at  the  points  of 
intersection  the  points  can  be  at  once  located,  provided 
the  pressure  curves  are  already  drawn,  otherwise  the 
corresponding  entropies  must  also  be  computed. 


74        THE  TEMPERATURE-ENTROPY   DIAGRAM. 

Another  method,  Fig.  29,  involving  less  computation, 
is  to  find  the  values  of  E  at  several  points  along  a  series 
of  isothermals  and  then  by  interpolation  to  connect 
corresponding  values  with  smooth  curves. 


FIG.  29. 

Fortunately  most  if  not  all  problems  occurring  in 
engineering  practice  can  be  solved  without  the  aid  of 
such  curves,  so  that  they  are  not  found  on  the 
ordinary  diagrams. 

It  is  thus  possible  to  construct  five  sets  of  curves, 
one  set  each  for  constant  pressure,  volume,  tempera- 
ture, entropy,  and  internal  energy  changes.  As  the 
intersection  of  any  two  of  these  gives  a  unique  location 


SUPERHEATED  VAPORS. 


75 


of  the  state  point,  it  follows  that  all  five  characteristics 
may  be  read  directly  from  the  diagram  as  soon  as  any 
two  are  known. 

Projection  of  any  Curve  from  the  pv-  to  the  T(f>- 
Plane. — There  is  no  graphical  method  of  procedure  so 
that  the  curve  must  be  plotted  point  by  point  by  deter- 
mining the  pressure  and  volume  at  these  points  and 
finding  the  intersections  of  the  same  pressure  and 
volume  curves  in  the  T^-plane.  The  curve  obtained 
by  connecting  these  points  will  be  an  approximate  t epres- 
entation  of  the  original.  This  method  must  be  adopted, 
for  example,  in  the  case  of  any  indicator  card  where 
there  is  superheated  steam  in  the  cylinder  after  cut-off. 


v  n      a'  I'i 

FlG.  30. 

Graphical    Method    of    Representing    AE   and   W.— 

Following  the  general  method  outlined  for  saturated 
steam  on  pp.  60-62,  we  may  represent  the  various  heat 
functions  of  any  state  point  (as  A,  Fig.  30)  in  the  super- 


70 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


heated  region.  Draw  through  A  the  constant  pressure 
curve  Aba  and  the  constant  volume  curve  Acd.  Draw 
through  b  the  constant  volume  curve  tv>.  Draw  through 
points  A,  b,  a,  and  g  the  isodynamic  curves  EA,  Ei,  Ea, 
and  Eg,  cutting  the  constant  volume  curve  VA  in  the 
points  A,  e,  h,  and  d,  respectively.  Then  we  have 

H    represented  by  area  under  gab  A, 

EA  "          "     "         "     dhcA,    or    by    the 

sum  of  the  areas  under  ga  and  under  h  c  A ; 
WgA  represented  by  area  Ogab  Acdno,or  by  the 

area  a' a  b  Achia'. 

In  Fig.  31  let  AB  represent  any  process  whatever 


6        c 
FIG.  31. 


a   9 


which  in  order  to  be  perfectly  general  is  assumed  to 
start  with  superheated  and  to  end  with  saturated 
vapor.  Draw  through  A  the  constant  volume  curve 


SUPERHEATED  VAPORS.  77 

v  and  through  B  the  isodynamic  EB,  intersecting  VA 
at  C.  Then  the  heat  rejected  during  the  change  AB 
is  represented  by  the  area  ABba,  the  decrease  in  internal 
energy,  EA  —  EB,  by  area  ADCca,  and  the  heat  equiva- 
lent of  the  work  performed  upon  the  substance  by  area 
ABbcCDA. 

To  obtain  the  external  work  during  any  change 
measure  the  area  under  the  curve  which  represents  the 
change  with  a  planimeter.  Then  note  by  reference  to 
the  isodynamic  lines  the  initial  and  final  values  o'f  the 
internal  energy.  Then  AW=Q—AAE. 


CHAPTER  V. 

THE     TEMPERATURE-ENTROPY     DIAGRAM     FOR    THE 
FLOW  OF  FLUIDS. 

IN  dealing  with  fluids  in  motion  when  the  velocity 
of  any  given  mass  varies  from  moment  to  moment  it 
becomes  necessary  to  introduce  a  term  for  the  kinetic 
energy  of  translation  into  the  mathematical  expression 
for  the  conservation  of  energy. 

Suppose  an  expansible  fluid  to  be  flowing  continually 
through  a  conduit  of  varying  cross-section  (Fig.  32). 


FIG.  32. 

If  no  external  work  is  performed  by  the  fluid  other 
than  to  push  itself  forward  we  know  that  all  the  energy 
carried  out  across  section  2  must  be  equal  to  that 
brought  in  across  section  1  plus  the  heat  received  by 
the  fluid  in  its  passage  from  1  to  2.  If  further  we 

assume  the  conduit  to  be  constructed  of  a  perfect  non- 
78 


FLOW  OF  FLUIDS.  79 

conductor  of  heat  then  the  process  is  adiabatic  and  there 
must  exist  the  following  energy  balance  per  pound, 


The   increase   in   kinetic   energy   between  the   two 
sections  is  therefore 


If  the  process  is  assumed  to  be  frictionless  and  there- 
fore in  the  thermodynamic  sense  reversible  (see  p.  xi 
of  Introduction)  the  expansion  occurs  at  constant 
entropy.  That  is,  the  specific  volume,  temperature  and 
internal  energy  are  found  for  any  given  pressure  from 
the  equation  for  frictionless  adiabatic  expansion, 

pvn=  constant, 
where  n  =  k=  1.405  for  air  and  other  diatomic  gases; 

n=  1.035  +  0.1002    for    wet    steam     (x=  initial 

quality)  ; 
n=  1.3  to  1.33  for  superheated  steam. 

If  then  the  pressure  and  volume  at  section  1  are 
represented  by  point  1  in  Fig.  33,  it  follows  that  the 
specific  volume  for  any  lower  pressure  may  be  found 
upon  the  constant  entropy  curve  through  1.  Thus  for 
pressure  p2  the  volume  is  v2.  This  diagram  thus  enables 


80 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


us  to  give  a  graphical  interpretation  to  the  expression 
for  the  increase  in  kinetic  energy  between  1  and  2. 
The  first  term  piVi  is  represented  by  the  rectangle  01, 
the  second  term  by  the  rectangle  02.  The  last  two 
terms  together  measure  the  decrease  in  internal  energy 
between  sections  1  and  2  and  are  represented  by  the 


FIG.  33. 

work  performed  during  this  expansion,  even  if  expended 
in  producing  self-acceleration.  Therefore  Ei—E2=J  pdv 

is  represented  by  the  area  under  the  curve  12.  The 
algebraic  sum  of  these  different  areas  shows  that  the 
increase  in  kinetic  energy  during  frictionless,  adiabatic 
flow  is  represented  by  the  area  between  the  expansion 
line  and  the  pressure  axis,  or 

72 


FLOW  OF  FLUIDS. 


81 


This  discussion,  which  up  to  this  point  has  been  per- 
fectly general,  must  now  be  carried  on  independently 
for  each  substance. 

Perfect  Gases. — If  it  is  possible  to  project  each  point 
of  the  pv-plane  uniquely  into  the  T^-plane  it  follows 
that  a  given  area  in  the  one  plane  can  be  transferred 
to  the  other  by  simply  determining  its  boundary  curves 


FIG.  34. 


in  the  two  planes.  The  area  in  the  T^-plane  will  then 
be  the  heat  equivalent  of  the  corresponding  area  in  the 
pu-plane. 

Let  12,  Fig.  34,  be  the  TV-projection  of  curve  12  in 
Fig.  33.  The  constant-pressure  curves  through  1  and  2 
will  intersect  at  infinity  with  the  curve  of  zero  volume 
so  that  the  cross-hatched  area  will  represent  the  heat 
equivalent  of  the  increase  of  kinetic  energy. 

A  slight  transformation  of  the  above  formula  makes 


82        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

it  possible  to  replace  the  infinite  diagram  by  an  equiva- 
lent finite  one  (Fig.  35).     Thus  from 


V2 


it  follows  that 


V* 


V2 

A-A-z-  is  thus  represented  by  the  area  under  the 

constant  pressure  curve  between  the  upper  and  lower 


FIG.  35. 

temperature  levels.  Thus  in  expanding  from  1  to  2 
the  heat  equivalent  of  the  increase  in  kinetic  energy  is 
shown  by  the  area  under  12'.  If  the  expansion  is  con- 
tinued to  3  the  further  increase  in  kinetic  energy  is 
shown  under  2'3',  while  the  total  amount  is  shown 
under  13'.  The  same  result  might  have  been  obtained 
from  Fig.  34  by  remembering  that  all  the  constant- 


FLOW  OF  FLUIDS.  83 

pressure  curves  for  a  perfect  gas  have  exactly  the  same 
form,  so  that  the  curve  2'oo  is  an  exact  reproduction 
of  2 oo  ,  and  therefore  the  total  area  under  1  minus  that 
under  2  leaves  the  area  under  12'. 

An  interesting  comparison  between  the  flow  of  a 
perfect  gas  and  the  velocity  of  a  freely  falling  body  can 
be  made  from  this  method  of  presentation.  Imagine 
the  expansion  to  continue  until  the  final  pressure 
becomes  zero,  when  the  volume  will  be  infinite.  The 
kinetic  energy  will  then  equal  cpTi.  This  represents 
the  total  energy  existing  in  space  at  1  due  to  the  presence 
of  the  pound  of  gas,  i.e.,  suppose  one  pound  of  air  at 
zero  temperature  could  have  been  inserted  into  this 
space  at  1  under  pressure  pi  and  then  heated  to  TI, 
the  total  energy  thus  intreduced  would  be  cp-T\.  If 
then  we  represent  this  total  heat  by  H,  we  notice  that 
the  increase  in  kinetic  energy  is  equal  to  the  decrease 
in  total  heat  or 

~  =  778-4H,    whence     V2  =  V2g  •  778  •  AH    if    7i  =  0. 

It  is  thus  seen  that  the  "total  heat"  head  (Erzeu- 
gungswarme)  plays  the  same  role  in  the  acceleration  of 
fluid  flow  that  gravitational  head  does  in  the  case  of  a 
freely  falling  body. 

Saturated  and  Superheated  Vapors. — Before  project- 
ing into  the  !T<£-plane  draw  in  the  liquid  (a)  and  dry 
vapor  (s)  lines  (Fig.  36).  It  is  then  seen  that  the 


84        THE   TEMPERATURE-ENTROPY   DIAGRAM. 

expansion  line  may  be  wholly  in  (1)   the  saturated,  or 
(2)  the  superheated  region,  or  (3)  may  pass  from  the 


s\ 

d    a     e\A  A, 


FlG.  36. 


superheated  to  the  saturated  region.     That  part  of  the 
diagram  abed  bounded  by  the  liquid  line,  the  adiabatic 


FLOW  QF  FLUIDS.  85 

expansion,  and  the  two  pressure  curves,  may  be  at  once 
located  in  the  T(f>-p\ane  as  indicated  in  Fig.  37.  The 
little  rectangle  to  the  left  of  the  liquid  line  is  retained 
in  the  pt'-plane.  Area  abed  in  the  77^>-plane  is  seen 
to  be  equal  to  Hi—H2,  and  area  klmn,  expressed  in 

heat  units  is       Z£2    .    Therefore 


77 

i~ 


778 


Now  H  +       +E32o  is  the  total  energy  (Erzeugungs- 

warme)  existing  in  any  space  due  to  the  presence  of 
the  pound  of  substance  under  these  conditions.  The 
difference  in  this  total  space  energy  differs  by  the  amount 

^Pl~o       from  the  difference  of  total  heats  (#i-#2). 

7/o 

Thus  if  7i  =  0 


V2  =  V2g[778(H1  -  H2)  +  (Pl  -  p2}o\ 

Here  again  the  "  total  energy  "  plays  the  same  role 
as  it  does  with  the  perfect  gases,  and  thus  similar  to 
that  played  by  gravitational  heacLin  the  case  of  a  freely 
falling  body. 

We  can  now  at  once  write  out  the  formula  required 
for  each  of  the  three  special  cases. 


THE  TEMPERATURE-ENTROPY  DIAGRAM. 


V2 
(1)    A-4-  = 


(2)  =  qi  +  rl+cp(TSl-T1)-q2-r2 

-cp(TS2-T2)+A(pl-p2)a; 

(3) 


For  the  sake  of  those  who  may  prefer  analytical 
methods  to  graphical  ones  the  ordinary  manner  of 
deriving  these  formulae  will  be  given  next. 

Starting  from  the  fundamental  equation, 

72 

—  E2, 


we  must  substitute  in  each  case  the  special  values  for 
volume  and  internal  energy  and  then  collect  terms  . 

V2 

(1)    AJ  K-  =  Api  (xiUi  +  a)  -Ap2(x2u2  +  a)+  qi+Xipi—q2 


V2 

(2)   A4-- 


-q2-  X2p2  -  x2Ap2u2 


Sl  -  Tl)-q2-p2 
-Ap2u2-cP(TS2-T2)+A(pl  -p2)a. 


FLOW  OF  FLUIDS.  87 


72 

(3)  A4- 


—  ?2  -  ^2|02  +  Cp(TSl  -  TI)  -  Api  (Vi  -  Si) 

=  qi  +  pi  +  ApiUi  +  cp(TSl  -  TI)  —  q2-  X2p2 


If  we  consider  the  flow  through  such  conduit  or 
nozzle  to  be  part  of  a  continuous  cycle  of  operations, 
it  becomes  necessary  to  return  the  water  to  the  boiler, 
that  is,  to  pump  it  back  again  from  p2  to  p\,  the  work 
required  being  A(pl  —  p2)ff.  Hence  if  this  term  be 
subtracted  the  difference  will  represent  that  part  of 
the  increase  in  kinetic  energy  which  is  available  for 
external  work,  such  as  driving  the  rotor  of  a  turbine. 
Thus, 

V2 


Design  of  a  Turbine  Nozzle.  —  Suppose  now  we  wish 
to  design  a  nozzle  to  permit  the  flow  of  G  pounds  of 
steam  per  second.  At  any  cross-section  of  area  F 
the  necessary  and  sufficient  condition  for  continuity 
of  flow  is  that 


- 


- 


For  any  pressure  px  there  can  be  only  one  definite  value 
for  velocity  and  specific  volume,  viz.,  Vx  and  vx,  and 


88          THE   TEMPERATURE-ENTROPY  DIAGRAM. 

hence  a  unique  value  of  the  area  Fx.  The  value  of 
vx  may  be  read  directly  from  the  constant-volume 
curves;  the  value  of  Vx  must  either  be  computed 
from  the  above  formulie  or  else  the  area  cdAB 
measured  by  planimeter  and  Vx  computed  from  that. 


In  Fig.  38  let  a'b'  be  the  /^-projection  of  the  fric- 
tionless  adiabatic  flow  ab.  Let  MN  represent  the  rela- 
tive variations  of  specific  volume  and  velocity  during 
this  expansion.  Draw  the-  line  yy  parallel  to  Ov  and 
make  Oy  equal  to  G.  Then  at  any  point  of  the  expan- 
sion as  d,  the  volume  v^  =  mdr  and  the  velocity  V4=nd", 


FLOW  OF  FLUIDS.  89 

Draw  Od"  and  prolong  if  necessary  until  it  intersects  yy 
in  k.  From  the  similar  triangles  Oyk  and  Od"n  it 
follows  that 

yk  :yO  =  On  :  nd" 

yk=y0.^,=G.^=F.         .     • 

Conversely  to  find  the  pressure  which  would  be 
obtained  at  any  cross-section,  lay  off  yk  =  F,  draw  W> 
and  prolong  until  it  intersects  MN  in  d",  from  d"  drop 
the  perpendicular  d"n  and  project  dr  back  to  d. 

The  smallest  cross-section,  or  the  throat  of  the 
nozzle,  will  be  reached  when  Ok  is  tangent  to  MN, 
viz.,  Ok',  giving  the  value  Fthroat  =  yk'. 

As  Ok  cuts  the  vF-curve  between  K  and  N  the 
value  of  F  increases  rapidly  until  at  N  it  becomes  infi- 
nite, which  agrees  with  the  initial  assumption  that 
Fa  =  0.  It  is  to  be  noticed  that  the  throat  is  reached 
when  the  pressure  has  dropped  to  about  0.58  pa.  From 
this  point  on  the  nozzle  flares  indefinitely  as  long  as 
the  back  pressure  is  dropped. 

Constant  Heat  Curves. — The  frequency  with  which 

V2 
the   expression   Ad -^- =  AH -\- A(pi  -  p2}o    must     be 

t7 

evaluated  in  turbine  design,  and  the  inconvenience  of 
solving  for  the  final  quality  by  equating  the  entropies 
in  order  to  obtain  H2  has  made  it  advisable  to  plot 
total-energy  curves,  H  +  Apa=  constant;  In  the  range 


90         THE   TEMPERATURE-ENTROPY   DIAGRAM. 

of  conditions  usually  met  with  the  term  Apa  is  negli- 
gible, so  that  the  total-energy  curves  practically  coincide 
with  the  constant  heat  curves.  Care  must  be  taken, 
however,  in  cases  of  extremely  high  pressures  and  wet 
steam  to  be  sure  that  A  pa  is  really  negligible.  Thus 
at  the  upper  pressure  quoted  in  Peabody's  Steam 
Tables  for  hot  water, 

336  X  144  X.  016 
#=2  =  402.2  B.T.U.    and    Apo=—    —  ===  — 

=  0.995  B.T.U., 

or  H  differs  from  H+Apa  by  0.25  per  cent,  approxi- 
mately. 

Thus  we   have,   in  general,   the   following  relation 
existing  between  any  two  points  of  such  a  curve 


H  +  Apa  =  qi  +  Xir  i  +  Ap^a  =  q2  +  r2  +  cp(Ts  -  T2}  +  Ap2a, 
while  for  all  ordinary  conditions  reduces  to 


To  plot  the  curve  in  the  saturated  region  a  series  of 
values  for  x  from  x  =  -  -  must  be  computed  for  a 

sufficient  number  of  different  temperatures. 

In  the  superheated  region  we  must  determine  the 
points  of  intersection  of  the  desired  constant-heat  curve 
with  several  constant-pressure  curves  (Fig.  39),  by 
means  of  the  relation 


FLOW  OF  FLUIDS 


91 


As  cp  is  a  variable  a  few  trials  may  be  necessary  before 
the  correct  value  of  Ts  is  obtained.     Ts  once  known, 
the  point  can  be  at  once  located  upon  the  corresponding 
constant-pressure  curve. 
The  constant-heat  curves  once  located  on  the  dia- 


0.2       0..4       0.6       0.8       1.0       1.2       1.4 

FIG.  39. 


1.6       1.8      2.0       2.2      2.4  <f> 


gram  the  solution  of  nozzle  problems  becomes  very 
simple.    Given  any  reversible  adiabatic  expansion,  as 

72 


AB,  Fig.  39,  the  value  of  ^rp 


obtained  by  reading 
the  values  of  HA  and  HB  directly  from  the  total  heat 


curves. 


92        THE    TEMPERATURE-ENTROPY   DIAGRAM. 

Peabody's    Temperature-entropy    Tables. — It    is    at 

this  point  that  the  practical  value  and  great  convenience 
of  Peabody's  Entropy  Tables  become  manifest.  These 
tables  cover  that  portion  of  the  diagram  (Fig.  40) 


1.52       1.88       2.18 
FIG.  40. 


between  <£i  =  1.52  and  <£2  =  1.83  from  the  upper  limits 
of  our  knowledge  of  saturated  and  superheated  steam 
down  to  85°  F.  In  this  region  constant  entropy  lines 
are  drawn  for  each  0.01  of  a  unit  of  entropy.  These 


FLOW  OF    FLUIDS.  9.3 

lines  are  then  crossed  by  a  series  of  constant-pressure 
curves,  so  spaced  that  in  the  saturated  region  they  coin- 
cide with  isothermals  spaced  one  degree  apart  from  420° 
to  85°.  The  table  then  tabulates  for  us  along  each 
constant-entropy  line  opposite  each  pressure  curve 
the  corresponding  temperature  of  saturated  steam,  the 
quality  (i.e.,  either  the  degrees  superheat  at  constant 
pressure  or  the  value  of  x),  the  specific  volume  com- 
puted from  the  characteristic  equation  for  superheated 
steam  or  from  v  =  0.016  +  x(s  —  0.016)  as  the  case  may 
be,  and  finally  the  value  of  the  total  heat  H  =  q  +  r 
+  cp(Ts-T}  orH=q  +  xr. 

The  entropy  tables  are  then  to  be  used  in  exactly 
the  same  manner  as  the  T^-diagram.  Each  is  entered 
by  knowing  the  initial  temperature  and  pressure,  or 
temperature  and  quality.  At  this  point  the  values  of 
v  and  H  are  noted.  The  eye  then  runs  down  the 
constant-entropy  line  or  column  until  the  desired  back 
pressure  is  found.  At  this  point  the  quality,  specific 
volume,  and  total  heat  are  read. 

Design  of  Nozzle  for  Frictionless  Adiabatic  Flow. — 
To  illustrate  the  use  of  the  T^-diagram  or  of  the  entropy 
tables  let  us  find  the  throat  and  final  diameter  of  a 
nozzle  capable  of  delivering  10  H.P.  net  in  kinetic 
energy  at  the  final  section.  Assume  the  steam  to  be 
initially  under  150  Ibs.  absolute  per  square  inch  and 
superheated  100°  F.,  and  that  the  discharge  is  at 
atmospheric  pressure.  ,  - 


94         THE   TEMPERATURE-ENTROPY   DIAGRAM. 

The  nozzle  must  deliver  10X550  =  5500  ft.-lbs.  of 
kinetic  energy  per  second..  Assuming  the  steam  to 
be  initially  at  rest  the  kinetic  energy  delivered  per 
pound  of  steam  is 


-#2)  =778(1249.6  -1068.1) 
=  141,200  ft.-lbs.  per  second. 
The  amount  of  steam  required  per  second  is  therefore 


At  the  throat  or  minimum  cross-section  the  pressure 
will  have  dropped  to  about  0.58  of  its  initial  value  or 
87  Ibs.  At  the  throat,  therefore,  the  specific  volume 
will  be  5.319  cu.  ft.  and  the  total  heat  1199.7  B.T.U. 

Hence 


yJ  =  x/778X64.32(1249.6-1200.4)  =  1569  ft.  per  sec. 

and 

_,     Gvt    0.03895X5.319  |Qon 

Ft  =-pr  = T =0.0001320  sq.  ft. 


and  Dia*=0.1256  in. 

Similarly  at  the  exit  cross-section, 

Ve  =  V778 X  64.32(1249.6  - 1068.1)  =  3014  ft.  per  sec. 
and 


and  Diae=  0.2408  in. 


FLOW  OF  FLUIDS.  95 

This  computation,  as  well  as  the  graphical  method  on 
pages  82-84,  gives  no  idea  as  to  the  length  of  the  nozzle. 
However,  the  exit  cross-section  would  need  to  be  placed 
at  such  a  distance  from  the  throat  as  to  make  the  flare 
of  the  nozzle  agree  with  the  natural  flare  of  the  jet. 
The  portion  leading  up  to  the  throat  must  be  rounded 
off  into  a  smooth  surface. 

The  design  of  a  nozzle  for  an  actual  case,  showing 
how  to  allow  for  friction  losses,  will  be  taken  up 
later. 

Irreversible  Adiabatic  Processes. — So  far,  in  speaking 
of  adiabatic  lines,  reference  has  been  made  only  to 
reversible  processes ;  that  is,  the  expansion  was  friction- 
less  and  work  was  done  at  the  expense  of  the  internal 
energy  either  upon  a  piston  or  in  .imparting  kinetic 
energy  to  the  molecules  of  the  expanding  fluid.  Sup- 
pose now  that  the  adiabatic  expansion  occurs  through 
a  porous  plug  (as  in  Kelvin  and  Joule's  experiments 
with  gases)  so  arranged  that  as  soon  as  velocity  dV  is 
developed,  it  is  at  once  dissipated  through  friction  into 
heat  dQ,  which  is  returned  to  the  body  at  the  lower 
pressure  p—dp.  The  first  operation  is  reversible  and 
hence  isentropic,  the  latter  is  equivalent  to  the  addition 
of  the  heat  dQ  from  some  external  source  and  hence 

the  entropy  increases  by  the  amount  -™ . 

The  actual  operation,  therefore,  results  in  a  drop  of 
pressure  and  a  growth  of  entropy  without  any  increase 


96         THE    TEMPERATURE-ENTROPY  DIAGRAM. 

in  velocity.     Referring  once  more  to  the  fundamental 
equation  for  the  flow  of  fluids, 

V2 

—  E2, 


V2 
we  obtain,  since  ^-  =  0  the  result, 


as  the  necessary  relation  between  any  two  conditions 
of  the  fluid  for  such  a  completely  irreversible  process 
as  flow  without  increase  of  velocity  or  the  performance 
of  outside  work  other  than  that  required  to  crowd  the 
substance  into  a  new  space. 

Irreversible  Adiabatic  Expansion  of  a  Perfect  Gas.  — 
In  the  case  of  perfect  gas 

pv+E  =  constant 

reduces  to 

k 


— 

-L 


constant,  or  simply  pv  =  constant, 


i.e.,  the  adiabatic  process  representing  expansion  with 
complete  friction  loss  is  at  the  same  time  an  isothermal 
and  an  isodynamic  change.  An  adiabatic  process  is, 
therefore,  indeterminate  unless  specifically  defined; 
if  reversible  it  coincides  with  an  isentrope;  if  absolutely 
irreversible,  with  a  total  energy  curve  which,  in  the 
case  of  perfect  gases  is  also  an  isothermal;  for  all  other 


FLOW   OF  FLUIDS. 


97 


(See 


changes    it   occupies   an   intermediate   position, 
pp.  ix,  x  in  the  Introduction.) 

Let  AB,  Fig.  41,  represent  such  an  irreversible 
adiabatic  process.  An  entirely  new  interpretation  must 
be  given  to  the  !T<£-diagram  for  such  processes  as  this. 
The  area  under  the  curve  AB  no  longer  represents  heat 
added  from  external  sources  (nor  from  any  source),  as 
no  heat  whatever  has  entered  the  body.  An  isothermal 
expansion  of  the  gas  has,  however,  occurred  without 


FIG.  41. 

the  performance  of  external  work  with  the  result  that 
PB<PA  and  VB>VA. 

The  real  significance  of  the  change  becomes  apparent 
if  we  bear  in  mind  Lord  Kelvin's  statement  of  the 
second  law  of  thermodynamics  that  "it  is  impossible 
by  means  of  inanimate  material  agency  to  derive 
mechanical  effort  from  any  portion  of  matter  by  cool- 
ing it  below  the  temperature  of  the  coldest  of  surround- 
ing objects." 

Let  thz  dotted  line  (Figs.  41  and  42)  represent  the 
lowest  available  temperature,  i.e.,  that  of  the  atmos- 


98         THE  TEMPERATURE-ENTROPY  DIAGRAM. 

phere  in  gas-engine  work.  The  minimum  available 
pressure  in  actual  work  would  likewise  be  that  of  the 
atmosphere. 


FIG.  42. 


The  work  developed  during  isentropic  expansion  to 
atmospheric  pressure  would  be 


and 


And  since  PAVA  =  PBVB,  pa  =  pb  and  PA>PB  it  follows 
that  W Bb  <  WAa  by  the  amount 


Therefore 


k-l\pB/          k-l\p 


FLOW   OF   FLUIDS.  99 

Furthermore,  while  the  work  done  upon  the  substance 
as  it  enters  the  cylinder  or  nozzle  is  the  same  (PAVA  = 
PBVB),  the  work  required  to  exhaust  it  has  increased 
by  the  amount 

The  total  loss  in  power  resulting  from  the  irreversible 
operation  AB  is  therefore 


Cp  jfc-l/       l-fe 

A.  atm.  \ 

In  other  words,  although  the  temperature  was  not 
changed  and  although  no  heat  entered  or  left  the  body 
during  the  change,  AB,  nevertheless,  because  of  it,  heat 
to  the  amount  cp(Tb—Ta}  has  been  made  non-available 
for  actual  work. 

Theoretically  the  loss  in  availability  is  not  quite  so 
great.  It  is  possible  to  conceive  of  the  expansion  being 
carried  below  the  back  pressure  until  the  lowest  avail- 
able temperature  is  reached,  i.e.,  from  b  to  c.  Then  on 
the  return  stroke  the  charge  could  be  isothcrmally 
compressed  from  c  to  a  so  that  the  extra  work  bca  could 
be  gained  and  the  corresponding  heat  loss  avoided 
(see  Fig.  41).  The  total  heat  made  theoretically  non- 
available  would  therefore  bs  represented  by  the  area 
under  ac,  i.e.,  by  TJ^^^B—^A). 


100       THE    TEMPERATURE-ENTROPY  DIAGRAM. 


Suppose  as  a  further  illustration  that  the  operation 
AB  were  introduced  into  a  cycle  in  which  all  the  other 
operations  were  reversible  isothermals  and  adiabatics 
(Fig.  43).  The  heat  received  from  some  outside  source 
is  shown  by  the  area  under  the  reversible  isothermal  CA 
The  heat  rejected  is  that  shown  by  the  area  under  the 
reversible  isothermal  be,  while  that  rejected  when  the 
irreversible  process  AB  is  eliminated  is  shown  by  the 
area  under  ac.  The  heat  exhausted  during  the  cycle 


c'  a' 

FIG.  43. 


has  thus  been  increased  by  the  amount  abb' a',  which  is 
equal  to  the  temperature  of  exhaust  multiplied  by  the 
increase  in  entropy  during  the  irreversible  process  AB. 
Swinburne  generalizes  this  result  in  the  following  words : 
"  The  increase  of  entropy  multiplied  by  the  lowest 
temperature  available  gives  the  energy  that  either  has 
been  already  irrevocably  degraded  into  heat  during  the 
change  in  question,  or  must,  at  least,  be  degraded  into 
heat  in  bringing  the  working  substance  back  to  the 
standard  state.  ..."  (See  p.  xiii  of  Introduction.) 

Transmission  of    Compressed  Air  Through    Pipes. — 
Air  delivered  by  a  compressor  is  heated  above  the  tern- 


FLOW   OF  FLUIDS. 


101 


perature  of  the  atmosphere.  This  excess  of  tempera- 
ture is  soon  lost  by  radiation  in  the  pipe  line  and  from 
that  point  on  the  flow  is  practically  isothermal.  There 
are,  however,  friction  losses  which  result  in  a  drop  of 
pressure  throughout  the  line.  Both  of  these  processes 
are  irreversible  and  both  produce  loss  of  power — the 
first  by  a  direct  radiation  of  heat,  the  second  by  a  re- 


duction in  availability  of  the  energy  remaining  in  the 
air. 

In  Fig.  44  let  ab  represent  the  cooling  at  practically 
constant  pressure  in  the  first  part  of  the  pipe  line,  and 
be  represent  the  irreversible  adiabatic  expansion  caused 
by  friction.  If  the  air  operates  a  motor  it  cannot  be 
expanded  below  atmospheric  pressure.  Then  adhf 
represents  the  work  which  the  air  as  delivered  by  the 
compressor  could  have  developed  during  frictionless 
adiabatic  expansion,  while  cehg  represents  the  work 
which  the  air  as  finally  delivered  at  the  motor  is  capable 
of  producing  during  frictionless  adiabatic  expansion. 


102       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  total  loss  of  power  is  thus  shown  by  the  area 
adecgfa.  A  more  complete  analysis  of  this  problem 
will  be  given  in  the  chapter  on  air-compressors. 

Irreversible  Adiabatic  Expansion  of  Saturated  and  Super- 
heated Vapors. — In  the  case  of  saturated  and  super- 
heated vapors  the  condition  for  completely  irreversible 
flow  pv  +  E  =  constant  reduces  to  H+Apa  =  constant. 
The  irreversible  adiabatic  therefore  coincides  with 
the  constant  total-energy  curves  or  approximately  with 
the  curve  of  constant  total  heat.  Such  a  process  occurs 
whenever  steam  pressure  is  lowered  through  a  reducing 
valve,  when  the  pressure  drops  as  steam  passes  through 
the  admission  and  exhaust  ports  of  an  engine,  or  when 
the  pressure  drops  throughout  the  length  of  a  pipe  line. 
From  its  direct  application  in  reducing  valves  this  process 
is  technically  known  as  throttling  and  the  constant 
total-energy  curves  are  sometimes  spoken  of  as  throttling 
curves  (Drosselkurven).  Here,  as  in  perfect  gases,  the 
adiabatic  process  is  found  to  be  indefinite  unless  the 
law  of  friction  loss  is  specified.  When  there  is  no  friction 
and  part  of  the  total  energy  goes  into  work  or  kinetic 
energy  it  coincides  with  an  isentropic  process.  When 
the  friction  loss  is  complete  so  that  no  acceleration  occurs 
it  coincides  with  a  throttling  curve.  In  cases  where 
the  friction  loss  is  only  partial,  as  in  turbine  nozzles, 
it  occupies  some  intermediate  position. 

It  should  be  noticed  that  during  throttling  of  satu- 
rated steam  the  moisture  tends  to  evaporate  and  that 


FLOW    OF   FLUIDS. 


103 


if  the  steam  is  nearly  dry  it  may  even  become  super- 
heated during  the  process.  We  have  found  that  for 
perfect  gases  the  throttling  curve  represents  an  iso- 
thermal process;  therefore,  the  further  the  throttling 
curves  extend  into  the  superheated  region  the  more 
nearly  horizontal  they  become,  approaching  the  iso- 
thermal line  as  a  limiting  case. 

Loss  of  Availability  due  to  Throttling  of  Steam. — 
Suppose  a  pound  of  steam  to  have  undergone  an  irre- 


FIG.  45. 

versible  change  of  condition  AB,  Fig.  45.  In  order  to 
restore  the  steam  to  its  initial  condition  it  must  be 
compressed  and  heat  must  be  rejected.  If  it  were  com- 
pressed adiabatically  from  BtoC  and  then  isothermally 
from  C  to  A  there  would  be  rejected  an  amount  of 
heat  equal  to  the  area  under  AC.  Evidently  less  heat 
need  be  rejected  if  some  path  falling  inside  of  BCA 


104       THE   TEMPERATURE-ENTROPY  DIAGRAM. 

were  utilized.  Evidently,  BB'A'A  is  the  path  along 
which  the  minimum  amount  of  heat  would  be  exhausted" 
when  B'A'  represents  the  lowest  available  temperature. 
There  would  be  no  advantage  in  continuing  the  adia- 
batic  expansion  BB'  below  B',  because  the  steam  could 
not  be  compressed  isothermally  at  any  lower  tempera- 
ture than  2V,  i.e.,  before  heat  could  flow  out  it  would 
need  to  be  compressed  back  to  B'  again  adiabatically. 
The  heat  which  is  unavoidably  lost  in  restoring  the 
steam  from  B  to  A  is  therefore  equal  to  the  lowest 
available  temperature  multiplied  by  the  growth  of 
entropy  during  the  change  AB. 

Problem  1.  —  A  throttling-  valve  reduces  steam  pres- 
sure from  150  Ibs.  gauge  to  80  Ibs.  gauge.  The  steam 
initially  contained  1  per  cent,  moisture.  If  the  steam 
is  used  by  an  engine  running  at  2  Ibs.  absolute  back 
pressure,  find  the  loss  per  pound  of  steam  caused  by 
the  valve. 

We  have  (omitting  Apa), 


?i64.7-f  0.99r164.7  =  g94.7  +  r94>7 
or 

337.8  +0.99X856.9  =294.4  +890.7  +  cP^-323.9), 

whence  cp(t8  -323.9)  =  1.0  B. 

From  the  table  of  values  of  cp  we  obtain  325.7  as  the 
value  of  t,  which  satisfies  this  equation. 


FLOW  OF  FLUIDS.  105 

Therefore, 


=  0.5235  +  0.99X1.0381 
-1.5513 

T  T 

and     2  =  02  +  r  +  ei>loe    r 


70  r 

=  0.4696  +  1.  1369  +  .  56  X  2.303  Xlogiy  ™ 

/oo 

=  1.6079. 

The  temperature  corresponding  to  2  IDS.  aosolute  is 
126.2°  F.  The  loss  of  available  energy  caused  by  the 
reducing-  valve  per  pound  of  steam  is  therefore  equal  to 

T(<f>2-<f>i}=  (126.2  +  459.5)  (1.6079-1.5513) 
=  33.2  B.T.U. 

Problem  2.  —  A  line  of  pipe  delivers  Wi  pounds  of 
steam  per  hour.  By  means  of  traps  and  a  separator 
placed  jiist  above  the  throttle  w2  pounds  of  water  are 
removed  per  hour.  The  steam  is  thus  practically  dry 
at  both  ends  of  the  pipe.  The  pressure  drop  from 
boiler  to  throttle  is  pi—p2  pounds.  The  steam-engine, 
using  the  Wi  pounds,  runs  with  p3  pounds  back  pressure. 
Find  the  total  waste  produced  by  transmission  through 
the  pipe. 

Assume  that  the  hot  water  is  returned  to  the  boiler 


100       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

at  the  throttle  temperature.    The  radiation  loss  (see 
Fig.  46)  therefore  equals 

w2(H1-q2)  +  w1(H1-H2')  B.T.U.  per  hour. 

Besides  this  direct  loss  of  heat  by  radiation  there  is  a 
further  indirect  loss  due  to  the  increase  in  entropy 
of  the  main  body  of  steam.  This  represents  the  extra 


*, 

J-    U 

T 

p,  /        \ 

/            4- 

7 

\ 

"/ 

\ 

/». 

ytr-?* 

\ 

v*,-^ 

/     t__ 

\ 

(-) 

Fiu.  40. 


heat  given  up  to  the  cooling  water  of  the  condenser 
during  exhaust  and  equals 


-^i)  B.T.U.  per  hour. 
The  total  loss  due  to  pipe  line  is 

-32)  +wi  (H^  H2)  +wi  -  T3(fa-  <£i)B.T.U.  per  hr. 
The  heat  supplied  to  the  water  in  the  boiler  is 

Wi(Hi-qd+w2(Hi-q2')  B.T.U.  per  hour. 


FLOW    OF  FLUIDS.  107 

The  efficiency  of  transmission 

Total  heat  supplied  — total  loss 
Total  heat  supplied 

Peabody  Calorimeter. — The  adiabatic  expansion  with 
increase  of  kinetic  energy  prevented  by  friction  finds 
a  very  simple  and  valuable  application  in  the  Peabody 
throttling  calorimeter.  The  calorimeter  is  a  small 
expansion  chamber  connected  to  the  steam  main  by 
means  of  a  small  pipe  supplied  with  a  valve.  A  second 
somewhat  larger  pipe,  also  containing  a  valve,  exhausts 
the  chamber  to  the  atmosphere  or  any  vacuum  space. 
The  whole  instrument  is  heavily  lagged  to  minimize  the 
radiation  and  make  the  process  adiabatic.  The  supply- 
pipe  sometimes  contains  a  standard  orifice  for  measuring 
the  steam  passing  through  per  hour.  Steam-pressure 
gauges  must  be  attached  to  the  steam  main  and  the 
calorimeter  chambers.  The  latter  must  also  contain  a 
thermometer  cup  to  permit  of  temperature  readings. 
To  operate  the  instrument  the  exhaust  and  admission 
valves  are  opened  wide  and  the  exhaust-valve  after- 
wards so  adjusted  as  to  produce  any  suitable  low 
pressure  in  the  calorimeter.  The  instrument  is  ready 
for  use  after  the  readings  become  constant  and  the 
thermometer  shows  a  minimum  of  about  10  degrees 
superheat.  An  excessive  amount  of  superheat  is  not 
advisable. 

The  action,  Fig.  47,  is  as  follows :  The  steam  expands 


108       THE    TEMPERATURE-ENTROPY  DIAGRAM. 


through  the  orifice  with  very  little  friction  loss,  the 
pressure  dropping  from  b  to  about  0.58'  of  its  initial 
value  at  a.  The  steam  as  it  leaves  the  nozzle  has  thus 
acquired  kinetic  energy  of  approximately  the  amount 
(Hb  —  Ha)  =  HPb—H58Pb.  The  jet  on  entering  the  lower 
pressure  of  the  calorimeter  chamber  expands  in  all 
directions,  eddies  are  set  up  and  the  kinetic  energy, 

/         Steam  pipe 


100 


H6— H0 


FIG   47. 

dissipated  by  friction,  is  restored  to  the  steam  as  heat 
and  thus  serves  to  evaporate  moisture.  The  cross- 
section  of  the  calorimeter  is  so  large  that  the  velocity 
of  the  steam  through  it  is  practically  the  same  as  in  the 
steam  main.  The  ultimate  change  in  kinetic  energy 
is  therefore  practically  zero,  and  as  no  external  work 
is  done  and  no  heat  lost  or  received  the  initial  condition 
of  the  steam  in  the  main  and  the  final  condition  in  the 


FLOW   OF   FLUIDS.  109 

calorimeter  chamber  must  represent  two  points  upon 
the  same  throttling-curve.  The  actual  path  of  the 
steam,  however,  is  not  down  the  curve  be  (Fig.  47), 
but  more  probably  down  some  such  path  as  bac,  the 
part  ac  being  indeterminate,  as  the  steam  is  not  in  a 
homogeneous  state.  Between  the  initial  and  final  con- 
ditions of  the  steam  we  have  the  simple  relation 

q_B  +  xrB  +  ApBa=qc+rc+cp(Ts-  Tc 

whence 

Hc-qs-A  (pB  - 


TB 
or  omitting  A(ps  —  pc)<?, 

Hc—qB 

x=  --  • 
TB 

The  application  of  the  Peabody  calorimeter  depends 
upon  the  possibility  of  superheating  the  steam  by 
throttling.  Thus  suppose  the  minimum  available  pres- 
sure is  that  indicated  in  the  diagram  (Fig.  48),  then 
steam  at  pressure  ps  and  of  the  quality  1,  2,  3,  or  4 
could  have  this  quality  determined  by  the  calorimeter, 
because  the  throttling  curves  through  these  points  in- 
tersect the  curve  pc  =  c  in  the  superheated  region,  so  that 
Hc  =  qc  +  rc  +  Cp(Ta  —  Tc)  is  known,  while  steam  of  the 
quality  5,  6,  7  could  not  have  its  quality  determined, 
because  the  throttle  curves  through  5,  6,  7,  .  .  .  ,  inter- 
sect the  lowest  available  pressure  curve  in  the  satu- 


110      THE    TEMPERATURE-ENTROPY   DIAGRAM. 

rated  region,  so  that  Hc=qc+Xcrc  is   unknown.    The 
equation  reads 


and  as  it  contains  two  unknowns  is  indeterminate,  so 
that  the  calorimeter  does  not  give  the  desired  informa- 
tion. 

It  is  evident  that  by  attaching  the  calorimeter  to  a 
vacuum  its  range  of  applicability  may  be  increased 


FIG.  48. 

and  further  that  the  higher  the  steam  pressure  the  greater 
the  amount  of  moisture  which  the  instrument  can  meas- 
ure. 

Specific  Heat  of  Superheated  Steam.— a.  By  Thrott- 
ing. — The  same  equation  which  permits  of  the  deter- 
mination of  the  value  of  x  by  assuming  cp  to  be  known 
may  of  course  be  used  the  other  way  around  and  start- 
ing with  a  known  value  of  x  permit  the  mean  value  of 
Cp  to  be  calculated.  This  method  was  used  by  Grindley  * 

*  Trans.  Royal  Soc.,  vol.  194,  sec.  A,  1900. 


FLOW  OF   FLUIDS.  Ill 

and  Griessman.*  Both  started  with  what  they  con- 
sidered to  be  dry  steam,  and  permitted  it  to  expand 
along  a  throttling  curve  into  the  superheated  region,  the 
one  used  an  orifice  in  a  glass  plate,  the  other  a  porous 
plug  of  canvas  washers  surrounded  by  wood.  Their  re- 
sults as  well  as  those  obtained  by  others  f  do  not  agree 
among  themselves  nor  with  those  obtained  in  other  ways. 

The  explanation  of  the  discrepancy  between  the 
results  of  different  investigators  using  the  throttling 
method  lies  undoubtedly  chiefly  in  the  difficulty  of 
knowing  exactly  the  quality  of  the  steam  at  the  initial 
conditions. 

The  discrepancies  between  the  results  obtained  by 
this  method  and  other  methods  undoubtedly  lies  in  the 
inaccuracies  of  the  steam  tables.  Thus  in  the  equa- 
tion 


q2+r2-qi  ranges  from  900-1200  B.T  U.,  so  that  an 
error  of  several  B.T.U.  in  the  determination  of  the 
various  quantities  and  a  large  error  in  the  assumption 
of  the  value  of  cp  would  have  but  small  effect  upon  the 
value  of  xi.  But  in  the  equation 


*  Zeitschrift  des  Vereins  Deutscher  Ingenieure,  vol.  47,  pp.  1852- 
1880. 

t  Theses,  M.  I.  T.,  1904,  1905,  1906. 


112       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

the  numerator  is  a  small  quantity  and  an  error  of  a 
small  amount  which  would  not  be  appreciable  in  the 
individual  terms  produces  a  large  error  in  the  value 
of  cp.  Furthermore,  as  TI  is  in  the  neighborhood  of 
1000  B.T.U.  a  very  slight  error  in  the  determination 
of  x\  would  produce  a  large  percentage  error  in  cp. 

It  would  seem  therefore  that  this  throttling  method 
cannot  give  accurate  results  until  the  values  in  the 
steam  tables  have  been  more  accurately  determined 
and  until  the  investigator  can  be  absolutely  sure  that 
the  steam  is  dry  initially. 

b.  By  Superheating  at  Constant  Pressure. — A  method 
used  by  Knoblauch  and  others,  and  which  avoids  the 
inaccuracies  of  the  throttling  method,  is  to  pass  steam 
already  somewhat  superheated  through  a  bath  elec- 
trically heated,  thereby  raising  its  temperature  at 
constant  pressure.  If  the  electrical  energy  Q  (corrected 
for  the  heat  absorbed  by  the  bath  and  metal  of  the 
calorimeter  and  that  lost  by  radiation)  required  for 
heating  w  pounds  and  the  corresponding  increase  in 
temperature  T2  —  Ti  are  known,  then  the  mean  value 

of  cp  for  the  given  conditions  is  —T^ ^rr- 

w(i  i  —  l  2) 

In  actual  operation  the  experiment  is  not  entirely 
so  simple  as  this.  The  steam  in  passing  through  the 
coil  of  pipe  in  the  bath  suffers  a  drop  of  pressure  p\  —  p2, 
caused  by  the  throttling  action  of  the  pipe  friction. 
The  friction  loss  occasions  a  drop  of  pressure  Ti  —  Tx, 


FLOW   OF  FLUIDS. 
Q 


113 


so  that  the  heat  added  -  raises  the  temperature  by  the 

amount  T2-TX  instead  of  only  r2-7Y  Now  the 
actual  change  is  not  along  the  broken  path  AGE  (Fig. 
49)  but  along  some  indeterminate  path  AB,  therefore 


.Is.... 


FIG.  49. 

the  throttling  action  does  not  all  occur  along  the  con- 
stant total-energy  curve  AC,  but  it  is  occurring  along  the 
whole  series  of  throttling  curves  between  A  and  B  where 
the  drop  in  temperature  is  not  so  great  as  along  AC. 
It  is  therefore  probable  that  the  total  drop  of  tempera- 
ture caused  by  throttling  is  not  Ti  —  Tx  but  somewhat 
smaller  than  that.  But  if  the  drop  of  pressure  pi-p2 


114       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

is  small  and  the  increase  in  temperature  T2  —  7\  large, 
the  term  7\  —  Tx  is  but  a  small  correction  factor,  and  a 
small  error  in  it  has  but  little  effect  upon  cp. 

How  is  Ti-Tx  to  be  determined?  The  law  of 
variation  of  cp  is  unknown  and  to  be  investigated. 
It  is  known  roughly  that  cp  varies  along  a  constant- 
pressure  curve,  decreasing  with  increase  of  tempera- 
ture. Therefore  the  mean  value  of  cp  for  T\  —  TX  is 
greater  than  the  mean  value  for  T2  —  Ti.  It  is  next 
assumed  that  if  the  throttling  curve  DF  is  drawn  that 
TE-TF  =  T1-TX.  This  is  not  far  from  true  as  the 
slope  of  the  curves  is  nearly  the  same.  Now  QGF= 
HD-  EG  and  QGE=cP(TD-TG),  whence 


Hp—HG 
cp 


and  therefore 

TI-TX=TE-TF=TD-TG- 


Introducing  this  correction  in  the  original  equation 

gives 

Q 


or        w(T2-Tl  +  TD-Tc)cP-w(HD-HG)=Q, 
whence 

~  +  HD-HG          -+HD-HG 


FLOW  OF  FLUIDS.  115 

or  since  HD-HG  =  0.305(tD-tG), 


Take  as  an  illustration  test  No.  2  of  the  experiments 
of  Knoblauch  and  Jakob, 

p!  =  29.71bs.          p2=27.71bs. 
=  249.8  <G=  245.8 


w=96.15lbs.         Q  =  3758B.T.U.        -=39.08 

39.08  +  0.305(249.8  -  245.8)     39.08  +  1.22 
Cp~  41  1.6  -329.9  +  249.8-245.8"   81.7  +  4.0 


It  is  true  that  the  inaccuracies  of  Regnault's  deter- 
mination of  the  total  heat  of  dry  steam  enter  into  the 
result,  but  whereas  in  the  throttling  method  they  affect 
the  entire  result,  here  they  enter  only  as  a  minor 
variation  in  a  small  corrective  term. 

The  apparent  mean  value  of  the  specific  heat  as 
obtained  from  the  direct  observations  is 


which  differs  is  this  case  by  about  1.7  per  cent,  from 
the  more  probable  value.    It  is  probable  that  after 


116       THE    TEMPERATURE-E\TROPY  DIAGRAM. 

the  first  approximation  of  cp  has  been  determined  for 
the  entire  range  of  experiments  a  second  and  more 
accurate  determination  of  T±  —  TX  could  be  made, 
giving  a  final  value  of  cp  as  accurate  as  the  original 
data. 

The  Total  Heat  of  Dry  Saturated  Steam.— In  dis- 
cussing the  throttling  experiments  of  Grindley  and 
Griessman  it  was  shown  that  the  discrepancies  in  the 
results  were  due  to  inaccuracies  in  the  existing  steam 
tables  rather  than  in  observational  shortcomings.  It 
remained  for  Dr.  Harvey  N.  Davis*  of  Harvard  to 
bring  order  out  of  chaos  and  to  point  the  way  through  a 
mass  of  apparently  conflicting  data  and  thus  to  lead  to 
the  establishment  of  new  steam  tables  in  which  all  the 
fundamental  values,  based  upon  different  experiments, 
are  found,  when  connected  by  Clapyron's  equation 

r  dp 

T'=Au-aT' 

to  check  up  with  a  maximum  deviation  not  exceeding 
0.2  of  one  per  cent.f 

Briefly,  Dr.  Davis  proceeded  as  follows:  Tho  values  of 
Cp  obtained  by  Knoblauch  and  Jacob  are  taken  as  rep- 
resenting the  best  attainable  data.  These  values  are 
then  substituted  in  the  throttling  observations  of 

*  Trans.  A.  S.  M.  E.,  1908,  pp.  741  to  774. 

t  Peabody,  "Steam  and  Entropy  Tables,"  p.  15. 


FLOW  OF  FLUIDS. 


117 


Grindley,  Griessman,  and  Peake*  and  establish  the 
difference  in  value  between  the  total  heats  at  two  dif- 
ferent temperatures,  as  Hi—H2=cP-4T2.  Thus  sup- 
pose that  observations  of  pressure  and  temperature 
have  been  taken  at  points  A,  B,  C,  D,  E  on  the  throt- 


FIG.  49a. 

tling  curve  AE,  Fig.  49a.  Then  there  exists  between 
the  points  A,  B',  C',  />',  and  Er  on  the  dry  saturated 
curve  these  differences : 

HA-HB,=cp(TB~TB,} 
HA-Hc,=cp(Tc-Tc,} 
HA-HD,=cp(TD-TD,\. 


*  Peake,  Proc.  Roy.  Soc.   (London),  Series  A,  vol.  76  (1905), 
pp. 185-205. 


118        THE  TEMPERATURE-ENTROPY   DIAGRAM. 

The  least  accurate  data  in  these,  as  with  all  other 
experiments  with  steam,  are  at  the  point  A,  due  to  the 
uncertainty  as  to  its  condition.  The  point  A  may, 
however,  be  eliminated  from  the  observations  by  com- 
bining the  above  equations  as  follows : 

HB,-Hc,=cp(Tc  -TC'}-cp(TB  -TV), 
HB>  —HD>  =  cp(TD  —  TV)  —  cp(TB  —  7V)> 
HB,  -HE^cp(TE-TE>]  -Cp(TB  -  TV),  etc. 

Plotting  AR  and  T  as  rectangular  coordinates,  let 
the  line  HB  represent  the  total  heat  of  the  throttling 


HE' 


FIG.  496. 

curve  AE,  while  the  vertical  lines  BE',  CC',  DD',  etc., 
represent  the  heat  of  superheat  at  the  respective  con- 
ditions B,  C,  D,  E,  and  then  the  points  B',  C',  D' ',  and 
E'  will  represent  the  total  heat  of  dry  steam  at  the 
respective  temperatures.  This  avoids  the  necessity 
for  the  determination  of  the  temperature  of  point  A. 
(Fig.  496.) 
B'}  Cf,  D',  E'  thus  represents  a  portion  of.  the  dry 


FLOW   OF  FLUIDS.  119 

saturated  steam  curve  with  the  temperature  of  the 
points  determined  absolutely  while  the  total  heats  are 
determined  only  relatively  as  the  value  of  HA  and 
therefore  of  HB  is  unknown.  Grindley's  data  gave 
seven  fragments  of  the  saturated  curve,  Griessman's 
data  eleven  and  Peake's  six. 

By  skillful  combination  of  these  twenty-four  fragments 
Davis  obtained  the  curve  showing  the  most  probable 
relation  between  the  total  heat  and  the  temperature 
of  saturated  steam  from  £-212°  F  to  ^  =  400°  F. 

This  curve  is  represented  within  the  limits  of  experi- 
mental accuracy  by  the  second  degree  equation, 

ff  =  ff212+ 0.3745(^-212)  +0.000550(f-212)2. 

For  #212  Davis  takes  115C.3  B.T.U.,  this  being  the 
mean  of  the  values  obtained  by  Henning*  and  Jolyf. 
Peabody,  however,  uses  1150  B.T.U.,  which  is  the  result 
obtained  from  Henning's  formula. 

Flow  through  a  Nozzle. — In  the  case  of  the  flow 
through  an  actual  nozzle  the  operation  is  not  rever- 
sible. Heat  is  lost  by  radiation;  heat  is  conducted 
through  the  metal  of  the  nozzle  from  the  higher  to  the 
lower  temperatures;  and  friction  occurs  in  varying 
amounts  hi  different  parts  of  the  nozzle.  The  first 
loss,  the  rejection  of  heat  as  heat,  decreases  the  entropy 

*  Wied.  Ann.,  IV,  vol.  21  (1906),  pp.  849-78. 
t  In  an  appendix  (p.  322)  to  paper  by  Griffiths,  Phil.  Trans. 
(London),  vol.  186  (1895),  pp.  261  et  seq. 


120     THE  TEMPERATURE-ENTROPY  DIAGRAM. 

of  the  fluid,  while  the  other  two  losses  both  increase 
its  entropy.  It  might  happen  that  these  opposing 
forces  just  balanced  and  then  the  expansion  would  be 
isentropic  but  not  reversible.  In  general,  however, 
the  radiation  loss  may  be  made  small,  so  that  the  opera- 
tion is  nearly  adiabatic,  but  with  increasing  entropy 


FIG.  50. 
due  to   conduction  along  the  nozzle  and  to  friction 


Thus  starting  from  a,  Fig.  50,  the  actual  expansion 
curve  will  lie  between  the  isentrope  ab  and  the  con- 
stant-heat curve  abf  in  some  such  position  as  ac.  The 
heat  theoretically  available  is  represented  by  abde. 
By  friction,  etc.,  the  portion  6cc161  has  been  returned 
to  the  substance  at  a  lower  temperature,  hence  the 
kinetic  energy  of  the  jet  at  the  exit  c  is  equal  to 

area  abde— area 


FLOW  OF  FLUIDS.  121 

This  loss  would  make  itself  noticeable  in  two  ways. 
Decreased  kinetic  energy  means  decreased  velocity, 
and  increased  entropy  means  increased  volume.  That 
is,  if  a  nozzle  were  constructed  from  the  dimensions 
necessary  to  give  frictionless  adiabatic  flow  and  drilled 
at  different  points  so  as  to  measure  the  pressure,  the 
observed  pressure  would  be  found  to  be  greater  at  any 
given  cross-section  than  the  pressure  for  frictionless 
flow.  Stodola's  experiments  show  this,  and  also  that 
the  loss  is  at  first  slight,  being  practically  negligible 
down  to  the  throat,  but  increasing  from  there  onward 
more  and  more  rapidly  as  the  velocity  increases.  That 
is,  the  curve  ac  would  at  first  closely  approximate  ab, 
but  lower  down  branch  off  more  and  more  toward  the 
right. 

As  soon  as  the  curve  ac  has  been  accurately  located 
it  can  be  projected  into  the  pv-pl&ne  and  the  corre- 
sponding areas  for  different  cross-sections  of  the  nozzle 
determined  in  the  manner  already  indicated  for  the 
ideal  case  of  frictionless  flow. 

Adiabatic  Expansion  with  Partial  Friction  Loss. — 
The  better  to  understand  the  phenomena  of  flow  in  a 
nozzle  let  us  discuss  the  flow  of  steam  in  a  non-con- 
ducting nozzle  in  which,  however,  there  is  the  usual 
friction  loss. 

Let  ab  (Fig.  51)  represent  an  isentropic  expansion, 
ai  a  constant-energy  expansion,  and  ac  an  adiabatic 
expansion  with  some  friction  loss,  all  from  the  same 


122        THE    TEMPERATURE-ENTROPY  DIAGRAM. 

initial  condition  a,  at  pressure  pi,  clown  to  the  same 
back  pressure  p%.  At  6  the  kinetic  energy  is  greater 
than  that  at  a  by  the  amount  abgk,  at  i  the  kinetic 


FIG.  51. 

energy  is  the  same  as  at  a,  and  at  c  it  possesses  a  value 
intermediate  between  that  at  b  and  i.    Thus 


and 


A- 


na—nc, 


whence  the  loss  of  kinetic  energy  caused  by  the  friction 
along  ac  is  shown  by  the  difference,  or 

Vf-Vf 
A ^ =Hc—Hb. 


FLOW  OF  FLUIDS. 


123 


That  is,  the  heat  equivalent  of  the  loss  of  kinetic  energy 
is  thus  represented  by  the  difference  in  the  total  heat 


FIG.  52. 

for  the  two  final  conditions,  or  by  the  area  under  the 
curve  be  in  the  7^-diagram,  Fig.  52. 

If  a  constant-heat  curve  be  drawn  through  c  until  it 
intersects  the  isentrope  ab  in  d  we  obtain 

Hd  —  Hb  =  Hc  —  Hb, 

so  that  the  loss  of  kinetic  energy  will  also  be  shown  by 
the  areas  bdef  plus  efgh  (Figs.  51  and  52).  Thus  in 
place  of  utilizing  the  total  available  drop  Ha  —  Hb, 
as  in  the  case  of  isentropic  flow,  the  actual  expansion  ac 
has  only  utilized  that  portion  Ha  —  Hd  which  would 
have  been  utilized  during  an  isentropic  expansion 
from  a  to  d. 
As  the  friction  loss  increases  the  expansion  line  ac 


124        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

would  move  more  and  more  to  the  right,  so  that  the 
utilized  heat  Ha  —  Hd  would  grow  continually  smaller, 
and  finally  in  the  limiting  case  where  "the  friction  loss 
is  complete  ac  coincides  with  ai  and  Ha  —  Hd  reduces 
to  zero. 

If  we  analyze  closely  such  an  irreversible  adiabatic 
expansion  as  ac,  we  notice  that  it  is  made  up  of  in- 
finitely small  transformations  of  heat  into  kinetic  energy 
and  of  kinetic  energy  back  into  heat,  and,  taken  in  that 
sense,  we  see  that  the  total  area  under  ac  can  be  inter- 
preted to  mean  the  heat  equivalent  of  the  friction  loss. 
But  we  have  already  found  that  the  loss  of  kinetic 
energy  is  shown  by  the  area  under  be,  so  the  total  fric- 
tion loss  exceeds  the  loss  of  kinetic  energy  by  the  amount 
represented  by  area  abc.  In  other  words,  all  of  the 
friction  loss  is  not  irretrievable,  because  although  the 
friction  loss  has  occurred  the  energy  has  nevertheless 
been  returned  as  heat  at  lower  temperature,  and  is 
therefore  still  capable  of  being  partially  turned  back 
into  work.  That  is,  at  any  intermediate  point  on  the 
line  ac  the  total  heat  is  greater  than  that  on  the  line  ab 
for  the  corresponding  pressure,  and  therefore  more 
work  is  theoretically  obtainable  from  it.  The  efficiency 
of  the  excess  heat  is,  however,  not  as  great  at  this  lower 
temperature  as  it  was  originally,  but  its  availability  is 
not  wholly  destroyed  until  the  expansion  is  carried 
down  to  the  lowest  possible  temperature. 

Design  of  a  Nozzle  for  Actual  Flow. — The  design  of 


FLOW  OF  FLUIDS.  125 

a  nozzle  for  actual  conditions  is  fundamentally  the 
same  as  the  case  discussed  on  pp.  87,  88,  except  that 
the  law  of  friction  loss  is  now  assumed  known  and  pro- 
vision made  for  the  resulting  decrease  in  velocity  and 
increase  in  specific  volume.  Thus  if  the  friction  loss  at 
any  given  section  is  /  per  cent,  the  available  kinetic 

energy  at  that  section  is  only  1  —  r^  parts  of  that 
available  for  frictionless  flow,  or 


whence  the  actual  velocity  is 


-(Hi-H2)  ft.  per  second. 


The  kinetic  energy  lost  by  friction  is  restored  as  part 
of  the  total  heat  and  results  simply  in  improving  the 
quality  of  the  steam.  Thus  if  the  quality  were  x2'  as 

ffTT    _  fj   \ 

the  result  of  isentropic  expansion,  the  heat  —  —  * 

J-UU 

would  by  further  evaporation  cause  an  increase  in  xy 
of  the  amount, 

,    f(H1-H2] 
100r2 

so  that  the  actual  value  of  the  quality  would  be 


120       THE  TEMPERATURE-ENTROPY  DIAGRAM 

The  actual  specific  volume  would  then  be 

v2  =  0.016  +x2(s2-  0.016). 
In  case  the  steam  were  superheated  at  the  end  of  the 

-T(TJ    _  TJ   \ 

isentropic  expansion  the  heat  —  —  would  produce 

1UU 

an  increase  in  superheat  of  the  amount, 

f(Hl-H2} 
100cp     ' 

and  the  final  temperature  would  be 


The  final  volume  V2  could  then  be  found  from  the 
equation 

pv=85.85T-0.25Qp, 

or  from  the  entropy  tables. 

If  the  weight  passing  the  section  per  second  is  G 
pounds  the  cross-sectional  area  is  given  by 


The  value  of  G  is  determined  by  knowing  the  power 
which  the  nozzle  must  develop  and  the  total  friction 
loss  at  the  exit  section.  Thus  for  n  horse-power  de- 


FLOW  OF  FLUID.  127 

livered  by  the  jet  and//  as  the  total  percentage  friction 
loss  at  the  final  section, 


and 

rcX550  nX550 

Or  = 


Problem. — Find  the  throat  and  final  diameters  of  a 
nozzle  to  develop  10  H.P.  in  the  issuing  jet,  assuming 
a  friction  loss  of  3  per  cent,  and  20  per  cent,  at  these 
sections  respectively.  The  steam  is  initially  at  150  Ibs. 
absolute  pressure  and  superheated  100°  F.,  and  the 
back  pressure  is  4  Ibs.  absolute. 

The  actual  kinetic  energy  of  the  issuing  jet  per  pound 
is 

^-==778X0.80X(1249.6-987.1)  =  163,200  ft.-lbs. 
whence 


7/=\/G4.32X  163,200  =  3240  ft.  per  second. 
and 

irv  v-  Ken 


or  equals 

0.03370X3600 
10 


-  Per  secon(l 


=  12.13  lbs.  per  H.P.  per  hour. 


128       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  value  of  ay  from  tne  tables  is  0.8617  and  the 
increase  in  x/  due  to  friction  loss  is 


and  the  actual  final  quality  is 

ay=  0.8617  +  0.0522  =  0.9139. 
The  final  specific  volume  is 

v/=  0.016  +  0.914  (90.4  -  .02) 

-82.7cu.ft. 
The  final  cross-sectional  area  is  therefore 

0.03370X82.7 
F'=     -^240-  ~ 

=  0.  1239  sq.  ins. 

The  final  diameter  is  therefore  0.3972  inch. 

The  theory  of  the  flow  of  fluids  shows  that  for  isen- 
tropic  expansion  the  pressure  in  the  throat  of  the 
nozzle  is  about  0.54  of  the  initial  pressure  for  super- 
heated steam  and  about  0.58  of  the  initial  pressure  for 
saturated  steam.  In  case  the  expansion  passed  from 
superheated  to  saturated  steam  before  the  throat  was 
reached  the  only  available  method  of  determining  the 

throat  pressure  would  be  to  plot  the  ratio  —  for  several 
values  of  p2  and  thus  determine  the  pressure  corre- 

y 

spending  to  the  maximum  value  of  —  • 


FLOW  OF  FLUIDS.  129 

In  this  problem  the.  steam  is  still  superheated  at 
the  throat,  so  that  the  throat  pressure  is  about 

0.54Xl50=811bs.  abs. 

The  actual  kinetic  energy  of  the  jet  at  the  throat 
per  pound  is 

7.2 

^- = 778 X 0.97(1249.6  - 1 194.5)  =  41590  ft.-lbs. 

whence 


V,=  V64.32X 41590  =  1635  ft.  per  second. 

The  superheat  at  the  end  of  isentropic  expansion 
amounts  to  23.6°  F.  This  is  further  increased  by  the 
friction  loss  by  the  amount 

0.03X55.1     1.653 

A 1 1  — = , 

CP  CP 

To  solve  this  we  neyd  to  know  the  momentary  value 
of  cP,  but  not  having  that  we  notice  in  the  entropy 
tables  that  the  total  heat  at  this  point  increases  7.8 
B.T.U.  for  14.7°  superheating.  This  gives 

_  7.8 
Cp~14.7 
whence 


130        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  actual  superheat  at  the  throat  =  26. 6°  F. 
By  cross  interpolation  in  the  tables  the  corresponding 
specific  volume  is  found  as 

vt  =  ^  (5.678 -  5.550)  +  5.609  =  5.636  cu.  ft. 

/.o 

The  cross-sectional  area  at  the  throat  is 

,-,     0.03370X5.636 
*'-     -1635 Xl44 

=  0.01673  sq.  ins. 

The  throat  diameter  is  therefore  0.1460  ins. 

Assuming  the  flare  between  throat  and  exit  to  be  a 
uniform  taper  of  one  in  ten  the  length  of  the  nozzle 
beyond  the  throat  would  be 

10(0.3972-0.1460)  =  2.51  inch. 


CHAPTER  VI. 

MOLLIER'S  TOTAL  ENERGY-ENTROPY  DIAGRAM. 

THE  important  role  played  by  the  total  energy, 
i=H  +  Apa=E+pv,  of  steam  in  the  discussion  of  the 
phenomena  of  flow  led  Mollier  *  to  construct  a  diagram 
using  i  and  $  as  the  coordinates.  In  that  portion  of 
the  diagram  abed  required  for  turbine  nozzles  Apo  is 
negligible,  so  that  i=H  practically,  and  the  diagram 
is  thus  sometimes  called  the  "total  heat-entropy  dia- 
gram." 

Description. — The  general  character  of  the  plot  and 
its  relation  to  the  ^-diagram  are  shown  in  Fig.  53, 
where  for  convenience  in  projection  from  the  T$-  into 
the  i<£-plot,  or  the  reverse,  the  pT-  and  pz-quadrants 
are  also  given.  Plot  first,  q  +  Apo  and  6,  and  q  +  r  +  Apa 

and  6  +  7F,  to  obtain  the  water  (w)  and  dry  steam  (s) 

lines.  The  isothermals  representing  vaporization  at 
constant  pressure  are  next  obtained  by  laying  off  the 
heat  of  the  liquid  and  the  total  heat  of  dry  steam, 

*  Neue  Diagramme  zur  technischen  Warmelehre,  von  Prof.  Dr. 
R.  Mollier,  Dresden.  Zeitsch.  d.  Ver.  Deutsch.  Ing.,  Bd.  48 
S.  271-274. 

131 


132        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Hw  and  Ha,  for  the  desired  pressure,  as  at  a  and  6  on 
the  #-axis,  finding  the  corresponding  points  on  the 


water  and  steam  lines,  a  and  b,  and  connecting  these 
points  by  the  straight  line  oh.    The  same  line  could 


MOLLIER'S  TOTAL  ENERGY-ENTROPY  DIAGRAM.  133 

of  course  be  determined  graphically  by  projecting 
upwards  the  points  a  and  6  from  the  T^-plot.  The 
"  x-lines  "  or  quality-lines  are  constructed  by  sub- 
dividing the  constant-pressure  lines  between  the  water 
and  steam  lines  into  any  desired  number  of  equal 
parts,  as  tenths  or  hundredths,  and  connecting  .  the 
corresponding  divisions  on  the  different  lines  by  smooth 
curves. 

To  continue  the  constant-pressure  curves  beyond  the 
dry-steam  line  into  the  superheated  region  we  plot  for 
each  pressure  the  set  of  values 


or,  starting  from  the  end  of  any  pressure  curve  at 
the  dry-steam  line,  lay  off  the  extra  values 

AH  =  cp(Ts-T)    and    ^  =  cploge^. 

The  accuracy  of  this  part  of  the  diagram  is  of  course 
limited  by  our  incomplete  knowledge  of  the  laws  of 
variation  of  cp  for  superheated  steam.  Thus  the  plots 
in  Stodola's  Steam  Turbines  are  based  upon  Regnault's 
old  value,  cp  =  0.48,  while  those  in  Thomas'  Steam 
Turbines  are  based  upon  the  larger  but  still  constant 
value,  cp  =  0.58. 

To  plot  the  isothermals  in  the  superheated  region, 
start  with  the  temperature  corresponding  to  any  pres- 
sure A  on  the  dry-steam  line,  then  the  heat  required  to 


134      THE  TEMPERATURE-ENTROPY  DIAGRAM. 

superheat  at  constant  pressure  to  this  same  temper- 
ture  from  any  lower  pressure  will  be  cp(Ta  —  Tc)  and 
the  point  B  may  then  be  located  on  the  curve  p  =  c 
by  laying  off  cp(TA-Tp}  B.T.U.  above  the  intersection 
of  p  =  c  with  the  dry-steam  line  (Fig.  54).  The  iso- 


FIG.  54. 

thermals,  provided  the  7^-plane  is  already  drawn,  may 
be  located  graphically  before  the  constant-pressure 
curves  are  drawn  by  projecting  the  value  of  <£  for  each 
value  of  CP(TB—TA)  directly  from  the  T<f>-p\a,ne. 

Reversible  Adiabatic  Processes.  —  For  frictionless  adia- 
batic  flow  between  the  two  pressures  pi  and  p2  the  kinetic 
energy  of  a  jet  increases  by  the  amount 


A- 


Enter  the  plot  at  the  point  p\x\  or  piTi,  according  as 
the  steam  is  saturated  or  superheated,  and  follow  down 


MOLLIER'S  'I'OTAL  ENERGY -ENTROPY  DIAGRAM.  135 

the  isentrope  thus  determined  until  the  constant- 
pressure  line  p2  is  reached.  The  vertical  distance 
between  the  initial  and  final  points  gives  at  once  i\  —  ii, 
or  the  heat  equivalent  of  the  kinetic  energy  of  the  jet. 
Assuming  the  initial  velocity  to  be  zero,  the  final 
velocity  corresponding  to  any  pressure,  is  given  by 


fJ.A--223.7V32; 

from  which  we  obtain  the  following  table : 

M  (in  B.T.U.)  VJl 


(in  ft.  per  sec.) 

0.01           0.1  22.37 

0.04           0.2  44.74 

0.09           0.3  67.1 

0.16           0.4  89.5 

0.25           0.5  111.9 

0.36           0.6  134.2 

0.49           0.7  156.6 

0.64           0.8  179.0 

0.81           0.9  201.3 

1.             1.  223.7 

4.             2.  447.4 

9.             3.  671. 

25.             5.  1119. 

64.             8.  1790. 

100.  10.  2237. 

144.  12.  2684. 

225.  15.  3355. 

324.  18.  4026. 

400.  20.  4474. 

441.  21.  4698. 

484.  22.  4921. 

625.  25.  5593. 


136       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


An  auxiliary  plot  (Fig.  55)  using  Ji  as  ordinates  and 
V  as  abscissae  permits  the  determination  of  interme- 
diate values,  and  forms  a  valuable  addition  not  only 
to  the  i'0-plot  but  to  the  entropy-tables  as  well. 


400 


300 


100 


GOO 
6000 


Velocity  in  feet  per  second 

FIG.  55. 


Two  plots  are  advisable,  drawn  to  different  scales,  one 
to  be  used  for  small  values  of  M,  the  other  for  large 
values. 

If  we  project  the  values  of  the  velocity  on  to  the 
J{-axis,  as  indicated  in  Fig.  55,  we  obtain  two  scales, 


MOLLIER'S  TOTAL  ENERGY-ENTROPY  DIAGRAM.  137 

*§     °§       I 


138       THE  TEMPERATURE-ENTROPY   DIAGRAM. 

one  showing  Ji,  the  other  V,  both  measured  from  the 
same  origin.  If  a  sufficient  number  of  intermediate 
values  of  V  are  indicated  the  scale  may  be  plotted  on  a 
separate  piece  of  paper,  and  can  then  be  applied  directly 
to  the  i^-diagram  by  placing  the  scale  parallel  to  the 
isentropic  curves  with  its  zero  at  the  initial  point  of  the 
expansion.  The  position  of  the  final  pressure  line  on 
the  scale  will  indicate  the  velocity  directly  in  feet  per 
second. 

Reduction  of  Pressure  by  Throttling. — Throttling 
curves  represent  processes  during  which  the  total  energy 
remains  constant,  and  these  are  therefore  represented 
by  horizontal  lines  in  the  i<f>-p\ot.  Such  a  plot  permits 
of  the  ready  determination  of  the  quality  of  steam  from 
the  calorimeter  observations.  Thus  if  A  (Fig.  56)  is 
the  point  determined  by  the  pressure  pc  and  tempera- 
ture tc  in  the  calorimeter,  proceed  from  A  horizontally 
to  the  left  until  the  curve  IA  intersects  the  constant- 
pressure  curve  corresponding  to  the  boiler  pressure  PB, 
say  at  B.  The  position  of  B  with  reference  to  the  x 
lines  gives  the  desired  quality  XB. 


CHAPTER  VII. 

THE  TEMPERATURE-ENTROPY  DIAGRAM  FOR  MIX- 
TURES (1)  OF  GASES,  (2)  OF  GASES  AND  VAPORS, 
AND  (3)  OF  VAPORS. 

Mixture  of  Perfect  Gases. — If  several  gases  be  mixed 
in  the  same  vessel,  the  pressure  cf  the  mixture  is  equal 
to  the  sum  of  the  pressures  which  the  gases  would 
exert  if  they  occupied  the  whole  space  separately.  This 
result  discovered  experimentally  by  Dalton  is  true  of 
course  only  so  long  as  the  molecules  do  not  sensibly 
obstruct  each  other.  It  may  be  assumed  to  hold  rigidly 
in  the  ideal  case  of  perfect  gases. 

For  any  given  mixture  undergoing  reversible  opera- 
tions it  must  also  be  assumed  that  heat  interchanges 
occur  instantaneously,  so  that  during  any  change  what- 
ever the  pressure  and  temperature  at  any  given  instant 
are  always  uniform  throughout.  The  common  tem- 
perature may  be  measured  directly,  and  if  the  weights 
of  the  different  gases  confined  in  the  given  space  are 
known  the  specific  volumes  may  be  calculated.  From 
these  two  quantities  the  respective  specific  pressures 
follow  at  once  from  the  characteristic  equations. 

If  the  mixture  is  composed  of  n  constituents  possessing 
the  weights  mi ...  mn,  the  specific  pressures  p\ . . .  p», 


140      THE   TEMPERATURE-ENTROPY  DIAGRAM. 

the  specific  heats  cVl .  .  .  cVn,  and  cPl .  .  .cPn,  respectively, 
the  values  of  the  specific  heats  of  the  mixture  are  given 

by 


and  cp= 

and  the  specific  pressure  by 

p=pi+p2  +  .  .  .  +  pn. 

To  apply  the  7^-analysis  to  any  process  occurring  in 
such  a  mixture  it  is  sufficient  to  treat  it  as  a  simple 
gas  possessing  the  specific  heats  cp  and  cv  and  obeying 
the  law  pv  =  RT,  where  R  is  determined  from  some  known 
values  of  temperature  and  volume.  This  is  exemplified 
in  gas-engine  work,  and  even  in  air-compressor  work: 
although  air  is  always  treated  as  a  unit  and  its  con- 
stituents never  considered. 

Mixture  of  Gases  and  Vapors.— Experiment  has  again 
shown  where  a  given  gas  and  liquid  are  chemically 
inactive  and  where  the  gas  is  not  physically  absorbed 
by  the  liquid,  that  when  contained  in  the  same  vessel 
the  liquid  evaporates  as  if  in  a  vacuum,  and  the  pressure 
of  the  vapor  is  the  same  whether  there  is  gas  in  the 
vessel  or  not.  Common  examples  of  such  mixtures  of 
interest  to  engineers  are  to  be  found  in  the  air-pumps 
of  steam-engine  condensers  and  in  compressors  pumping 
moist  air  or  those  cooled  by  water  injection. 


MIXTURES  OF  GASES  AND  VAPORS.       .       141 

Let  there  be  w  pounds  of  a  gas  per  pound  of  satu- 
rated vapor  in  a  given  mixture  and  let  it  be  desired  to 
trace  the  relative  changes  between  the  two  constituents 
as  the  mixture  undergoes  various  definite  changes. 

From  equation  (8),  page  15,  we  have  as  the  change 
in  entropy  of  w  pounds  of  a  perfect  gas  in  going  from 
any  condition  to  any  other  condition  the  expression, 

A$g  =  WCp  loge  ^T-w(Cp  —  Cv)  loge  —  • 

Also  the  change  in  entropy  between  any  two  points 
in  the  region  of  saturated  vapor  is  evidently  equal  to 
the  difference  in  the  total  entropies  of  the  two  points,  or 


fi 

v  rT          v\  --  m~  • 

i  2  1  1 

Hence  the  total  entropy  change  of  the  mixture  is 
given  by  the  sum 


Pit 

A  thermometer  will  give  the  common  temperatures 
TI  and  T2  and  the  aid  of  suitable  tables  (steam,  ammo- 
nia, etc.)  determines  the  values  of  the  various  heat 
quantities  referring  to  the  vapor.  Subtracting  from 
the  gauge  readings  the  vapor-pressures  corresponding  to 


142        THE    TEMPERATURE-ENTROPY   DIAGRAM. 

the  observed  temperatures  from  the  tables  leaves  the 
gas  pressures  pgi  and  pg2. 

There  exist  also  the  further  relations  between  the 
volumes  of  the  gas  and  vapor, 


whence 

wvgi— a 


x2 


U2 


RT1  RT2 

W  ----  a  w  ---  —  a 


U2 


Substituting  these  values  of  Xi  and  x2  in  the  expression 
for  4(j)M  gives 


+  WCp  loge  7fr-w(cp-  Cv]  loge  —  • 
*   1  Poi 

In  this  equation  —  provided  it  is4  possible  to  measure 
pressure  and  temperature  —  there  are  but  two  un- 
knowns 4(j>M  an(l  w-  Hence  if  w  be  known  J<j)M  may 
be  determined.  For  any  given  process  it  is  sometimes 
possible  to  find  the  relative  weights  of  gas  and  vapor. 
Let  us  now  consider  several  special  cases  where  the 
process  is  given. 

(1)  Isothermal  Change.  —  During  an  isothermal  pro- 
cess the  pressure  of  the  saturated  vapor  remains  constant 


MIXTURES  OF  GASES  AND  VAPORS.  143 

but  that  of  a  gas  will  decrease  as  heat  is  added  (expan- 
sion) so  that  the  total  pressure  of  the  mixture  will 
drop. 

For  such  a  change  T2  =  Ti,  62  =  di,  r2  =  ri,  u2  =  Ui,  so 
that  the  general  expression  for  A$M  reduces  to 


To  measure  the  heat  received  by  the  mixture  add  to 
the  increase  in  total  heat  of  the  vapor  the  heat  equiva- 
lent of  the  work  performed  by  the  gas,  thus 


Taking  the  values  of  x2  and  x\  from  page  142  we 
obtain 

_w_  RT2      a     w    RTi  a_ 

~Xl~u2     pgt  ~u2~u~1    pgi  u^ 


so  that  the  equation  for  heat  received  may  be  written 

Q=RTw-  [—-—1  -  AwRT  loge  £B. 
u  I?*    Pf,J  &  Pv 

Of  course  this  same  expression  can  be  obtained  di- 
rectly by  multiplying  A$M  by  T. 


144       THE  TEMPERATURE-ENTROPY  DIAGRAM. 
The  work  performed  is  given  by 

~ 


Since  the  intrinsic  energy  of  the  gas  does  not  change, 


As  all  the  above  expressions  involve  w  it  becomes 
necessary  to  determine  the  pounds  of  gas  present  in 
the  mixture  per  pound  of  vapor  before  they  can  be 
evaluated. 

Let  a  (Fig.  57)  represent  the  initial  state  of  the  vapor 
and  also  of  the  mixture.  Then  in  changing  from  a 
to  6  the  vapor  receives  the  heat  shown  by  the  area 
under  oh  and  increases  in  volume  from  a'  to  b',  while 
the  pressure  maintains  the  constant  value  pv.  The 
heat  received  by  the  gas  may  be  represented  by  some 
such  area  as  that  under  be,  so  that  the  total  area  under 
ac  represents  heat  received  by  the  mixture,  while  ab, 
be,  and  ac  represent  J<£v,  J(j>g,  and  A<}>M,  respectively. 
Lay  off  PMl  and  PMt  equal  to  the  initial  and  final  gage 
readings,  then  Aaf  and  Bbf  will  represent  the  pressure 


MIXTURES  OF  GASES  AND  VAPORS, 


145 


of  the  gas  and  AB  referred  to  a'~bf  as  the  axis  of 
volume  will  represent  the  pv-curve  of  the  gas,  while 
referred  to  Ov  as  the  zero  pressure  line  will  represent 
the  pu-curve  of  the  mixture. 


FIG.  57. 


(2)   Heating  or  Cooling  at  Constant   Volume.  —  This 
introduces  the  special  conditions, 


,     and 


RTl       RT2 

-  =  w—  — 
Poi  Po-i 


By  substitution  of  these  values,  the  general  expres- 
sion for  the  entropy  of  a  mixture  reduces  to 

T2 

a  e  ^ 


146        THE  TEMPERATURE-ENTROPY  DIAGRAM. 
The  heat  received  during  this  process  is  given  by 


The  change  of  internal  energy  is  of  course  equal  to  the 
heat  added. 

Here  again  the  value  of  w  must  be  determined  by 
suitable  measurements  before  the  expressions  can  be 
evaluated. 

(3)  Isentropic  Expansion  or  Compression  —  If  the 
mixture  undergoes  frictionless  adiabatic  expansion  it 
is  evident  that  this  might  occur  in  one  of  two  ways: 
either  (1)  the  entropies  of  both  the  vapor  and  the  gas 
remain  constant,  or  (2)  if  the  entropy  of  the  one 
varies  a  given  amount  that  of  the  other  varies  an  equal 
amount  in  the  opposite  direction.  Which  of  these  two 
possibilities  is  true  and  what  are  the  character  and 
magnitude  of  the  actual  change? 

The  fundamental  relation  now  becomes,  since  4<f)M  =  Q, 


I"  Rr2       Rn 

=  W     -  —  +  C 

Lu2pg,    ulpgi  i  pgi 


T2  TV  1 

Tf,  --  AR  loge  —    • 

Ti  fe  pgi] 


Provided  the  temperatures  T\  and  T2  may  both  be 
read  this  expression  contains  but  one  unknown,  w,  and 
thus  gives  directly  the  relative  weights  of  gas  and  vapor 


MIXTURES  OF  GASES  AXD  VAPORS. 


147 


during  such  a  change.  Whether  the  work  is  performed 
upon  a  piston  or  in  accelerating  the  mixture  itself  is 
non-essential. 

w  being  determined  the  quality  of  the  vapor  at  any 
temperature  T  is  given  by 

RT 


Having  found  w  and  the  quality  Zi  it  is  now  possible 
to  determine  the  value  of  x  at  any  number  of  tempera- 


FIG.  58. 


tures  by  taking  suitable  readings  of  pressure  and  tem- 
perature.    Let  ah  (Fig.  58)  be  the  7^-projection  repre- 


148       THE   TEMPERATURE-ENTROPY   DIAGRAM. 

senting  the  locus  of  such  a  series  of  points  for  a  pound 
of  steam.  Then  ac  must  represent  the  corresponding 
curve  for  the  w  pounds  of  air,  and  ad  will  represent  the 
isentropic  path  of  the  mixture.  Thus  the  energy  rejected 
by  the  steam  as  heat  must  be  received  by  the  gas  as 
heat  so  that  the  process  as  regards  its  surroundings  is 
adiabatic.  If  ac  represent  the  path  of  the  steam  ab 
will  represent  that  of  the  air. 

Let  AD  represent  the  pv-curve  of  the  mixture,  then 
Aa',  Db',  etc.,  will  represent  the  pressure  due  to  the 
air. 

In  order  that  J<£r  =  J<£a=0  it  is  necessary  that 


or  that 


w-- 


For  all  other  values  of  w, 

4^0, 
and  the  magnitude  may  be  found  from 

-  J0,  =  A$9  =  w  L  loge  ~  AR  loge  5*1  • 
*  i  P<n.-* 

The  Determination  of  the  Quality  of  Exhaust  Steam- 
— In  making  the  heat  balance  of  an  engine  test,  in  a 


MIXTURES  OF  GASES  AND  VAPORS.  149 

Hirn's  analysis,  etc.,  it  is  often  desirable  to  know  the 
quality  of  the  exhaust  steam  from  a  cylinder.  Direct 
measurements  are  usually  not  possible  with  a  throttling 
calorimeter  due  to  the  large  quantity  of  moisture 
present.  The  above  method  lends  itself  to  this  purpose, 
provided 

(1)  An  orifice  can  be  inserted  in  the  exhaust-pipe 
without  appreciably  changing  the  back  pressure; 

(2)  That  the  friction  loss  down  to  the  throat  of  the 
orifice  is  negligible,  and 

(3)  That    the    pressures    and    temperatures    in    the 
exhaust-pipe  and  in  the  throat  of  the  orifice  can  be 
accurately  determined.     This  will  necessitate  the  use 
of  accurate  manometers  and  of  thermo-electric  couples. 

For  the  special  case  of  steam  and  air  we  have 


W=' 


53.351  —  - 


(4)  Constant-pressure  Changes.  —  These  processes, 
although  difficult  to  analyze,  must  receive  attention, 
as  in  gas-turbine  work  it  has  been  suggested  that 
water  be  injected  into  gas  burning  at  constant  pressure, 
and  although  the  steam  eventually  becomes  super- 
heated the  mixture  is  initially  one  of  saturated  vapor 
and  gas. 


150        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

We  have  besides  the  general  condition 


r2 


r2 


the  further  relation 

PM  =  Po  +  PV  =  constant. 

pM  may  be  constant  provided  both  pv  and  pff  are 
constant  or  if  pv  increases  as  rapidly  as  pg  decreases. 
The  first  method  is  impossible  because  pv  cannot  be 


FIG.  59. 

constant  unless  T  remains  constant,  but  if  T  is  constant 
pg  decreases.  The  second  solution  is  therefore  the  only 
possible  one.  That  is,  the  pv-curves  of  the  vapor  and 
gas  must  be  some  such  curves  as  pv  and  pg  respectively 
in  Fig.  59,  where  pg+pv  =  constant. 
These  curves  fall  under  one  of  three  headings: 

(1)  pv  and  pg  are  both  straight  lines; 

(2)  pv  is  concave  and  pg  convex;  or 

(3)  pv  is  convex  and  pg  concave. 


MIXTURES  OF  GASES  AND  VAPORS.  151 

The  work  developed  by  the  mixture  is  of  course 
pM(v2  —  v\}.  The  work  developed  by  each  constituent 
may  be  determined,  provided  temperature  observations 
can  be  obtained  at  several  volumes,  as  the  simultaneous 
observations  of  p,  v,  and  T  serve  not  only  to  determine 
the  weight  of  gas  present  but  also  to  define  the  curve  pg. 
The  area  under  this  curve  represents  that  portion  of 
the  work  performed  by  the  gas  and  the  area  above  the 
curve  that  performed  by  the  vapor.  If  it  is  further 
possible  to  measure  the  vapor  separately  by  weighing 
the  liquid  before  it  is  fed  into  the  combustion  chamber 
or  by  condensation,  the  value  of  w  is  also  known.  It 
then  becomes  possible  to  find  J^,  A$v,  and  A$M. 

The  heat  required  for  the  change  is  given  by 


where  xi  and  x2  are  given  by  the  formulae  on  p.  142. 

Vapor-pressure  of  a  Liquid  Mixture.  —  The  pressure 
of  the  saturated  vapor  of  a  mixture  of  liquids  was 
investigated  by  Regnault.  The  mixed  vapors  were 
found  not  to  behave  in  general  like  a  mixture  of  gases 
as  regards  pressure.  Regnault  distinguished  three 
cases:  (1)  when  the  liquids  do  not  mix,  as  water  and 
benzene.  In  this  case  the  vapor-pressure  of  the  mix- 
ture is  equal  to  the  sum  of  the  vapor-pressures  of  the 
constituents.  (2)  When  the  liquids  mix  partially  or 
dissolve  each  other  to  a  limited  extent,  like  water  and 


152        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

ether.  In  this  case  the  vapor-pressure  of  the  mixture 
is  less  than  the  sum  of  the  pressures  of  the  constituents, 
or  even  less  than  one  of  them.  Thus  Regnault  found 

Temp.         Water-vapor  Press.  Ether.  Mixture. 

15.56°  C.  13.16  mm.  361.4  mm.  362.95  mm. 
33.08°  C.  27.58  mm.  711.6  mm.  710.02  mm. 

(3)  The  third  case  is  that  in  which  the  liquids  mix  in 
all  proportions.  In  this  case  the  diminution  of  the 
vapor-pressure  of  the  mixture  is  still  more  marked. 

According  to  the  experiments  of  Wiillner  the  vapor- 
pressure  of  any  given  mixture  bears  a  constant  ratio 
to  the  sum  of  the  vapor-pressures  of  the  constituents, 
at  least  when  the  liquids  are  mixed  in  nearly  equal 
proportions.  For  other  proportions  this  law  is  not 
quite  exact.  (Preston,  Theory  of  Heat,  p.  406). 

Mixtures  of  Liquids. — Only  that  class  in  which  the 
components  exert  their  full  individual  pressures  can 
be  submitted  to  a  general  thermodynamic  treatment. 
All  other  mixtures  must  be  treated  individually  as  special 
problems. 

It  is  assumed  that  the  mixture  is  of  the  same  tem- 
perature throughout  at  any  given  moment  and  that 
the  relative  weights  in  a  given  space  remain  constant. 
Furthermore,  that  each  substance  fills  the  entire  space. 

Heating  at  Constant  Pressure. — Suppose  such  a  mix- 
ture of  liquid  confined  in  a  cylinder  under  pressure  and 


MIXTURES  OF  GASES  AND  VAPORS.  153 

heated  from  32°  F.  until  it  begins  to  vaporize.     The 
heat  required  would  be  the  total  heat  of  the  liquids, 


rt  rt  rt  p* 

=  101  I    cidt  +  w2  Jc2dt  +  w3  \   c3dt  +  .  .  .  =  Sw?  Icdt. 

«y  oJ  «y  o^  fc/  o^  «yo- 


The  upper  temperature,  that  is  the  temperature  at 
which  vaporization  occurs,  is  determined  by  the  rela- 
tion, 


The  entropy  of  the  mixture  is  given  by 

2dT  r'cdT 


During  vaporization 

.  .  .  =  vol.  of  cylinder, 


so  that  the  quality  of  each  constituent  can  be  found 
from  the  equations 


The  heat  required  for  vaporization  is  therefore  equal  to 


The  entropy  is 


x\r\ 

l-m— 
1  I 


154        THE  TEMPERATURE-ENTROPY  DIAGRAM. 
The  external  work  of  vaporization  is 


and  the  internal  energy  of  vaporization  is 


But  each  substance  possesses  at  the  temperature  t 
its  own  specific  volume,  so  that  there  is  one  value  of 
wv,  say  WiVi,  which  will  be  smaller  than  all  the  other 
values  of  wv.  That  is,  this  particular  substance  will 
be  the  first  to  have  its  liquid  vaporized.  At  the  moment 
this  occurs 

Vol.  ^ 
whence 


xrM  = 


xr\  TI  r2  r3 

TJT)      =  Wl  TfT  +  W2X27fr  +  W3X37fr  +  ,  .  .  . 

T/u        TI  T-2  T3 


etc.,  etc. 

Any  further  increase  in  volume  at  this  temperature 
would  mean  superheating  of  Wi  with  consequent  drop 
of  pressure  pi,  that  is,  pM  would  decrease.  This,  how- 
ever, is  inconsistent  with  the  assumption  of  constant 
pressure,  so  that  the  temperature  would  begin  to  rise 


MIXTURES  OF  GASES   AND  VAPORS.  155 

again  in  such  a  manner  that  the  rise  in  pressure  of 
P2  +  P3  +  P±  +  -  ••  would  just  counterbalance  the  drop  in 
pressure  of  pi.  The  increasing  temperature  causes  Si, 
S2,  $3,  -  •  -  to  decrease,  and  this,  combined  with  the 
further  movement  of  the  piston,  will  cause  one  after 
another  of  the  components  to  become  superheated. 

During  this  stage,  when  saturated  and  superheated 
vapors  are  intermingled,  if  we  assume  the  total  pressure, 
the  temperature,  and  the  total  volume  to  be  known, 
it  is  possible  to  determine  for  each  substance  whether 

or  not 

<      vol. . 
£s. 

w  < 

vol. 
Those  that  give  — ->s  are  superheated  and  the 

others  saturated.  The  pressures  of  the  superheated 
vapors  may  be  found  from  their  respective  characteristic 
equations,  the  pressures  of  the  saturated  vapors  from 
the  vapor  tables. 

The  total  heat  of  such  a  mixture  would  no  longer  be 
equal  to  the  sum  of  the  total  heats  of  its  constituents, 
after  the  first  one  had  begun  to  superheat,  but  could  be 
found  by  adding  to  2wq  the  total  work  performed, 
ApM(T.wv—  ^wa],  and  the  increase  of  internal  energy  of 
each  component. 

The  total  entropy  would  be  • 


156        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

After  all  the  vapors  were  superheated  the  specific 
pressures  of  each  could  be  found  by  inserting  the  known 
values  of  T  and  v  in  the  various  characteristic  equa- 
tions. Then  there  would  exist  as  a  check  the  relation, 

PM  (observed)  =  (?1  +JP2+?3  +  •  •  •  )  (computed)- 

This  extra  condition  could,  if  necessary,  be  used  to 
determine  the  weight  of  one  of  the  constituents. 

The  entropy  of  the  mixture  of  superheated  vapors 
would  be 


The  heat  required  to  raise  the  mixture  at  constant 
pressure  from  liquid  at  32°  F.  to  superheated  vapors  of 
temperature  T  is  found  by  adding  the  increase  of  internal 
energy  to  the  external  work,  or 


-. .  .  +  APM(WV—  £0) 
=  ~S,wE+ApM(wv—  2<r). 

Isothermal  Expansion. — Starting  with  a  mixture  of 
saturated  vapors  the  quality  of  each  vapor  is  given  as 
before  by 

vol.  vol. 

<7i  0% 

Wi  W2 

Xl~      ui     '    Xz  ~      u2     ' ' " 


MIXTURES  OF  GASES  AND  VAPORS.  157 

and  the  pressure  of  the  mixture  by 


As  heat  is  added,  one  after  another  of  the  constituents 
will  become  superheated,  and  as  the  temperature  does 
not  change,  the  pressure  of  such  a  superheated  vapor 
will  then  decrease  as  the  volume  increases.  The  change 
of  entropy  can  thus  be  computed  separately  for  each 
component,  so  that 


and  the  heat  added  during  the  change  is 

Q-T-WM- 

The  external  work  performed  during  such  a  change  is 


CHAPTER  VIII. 

THE    TEMPERATURE-ENTROPY    DIAGRAM    APPLIED 
TO    HOT-AIR    ENGINES. 

THE  Carnot  cycle  in  the  T^-plane  is  always  a  rec- 
tangle, but  in  the  pf-plane  its  shape  depends  upon  the 
nature  of  the  working  substance.  For  perfect  gases 
the  isothermals  and  frictionless  adiabatics  have  nearly 
the  same  slope,  so  that  to  obtain  an  appreciable  work 
area  either  the  diameter  of  the  cylinder  or  the  length 
of  the  stroke  must  be  made  excessively  large.  That 
is,  the  excessive  size  and  weight  of  the  engine  combined 
with-  large  radiation  and  friction  losses  make  the  use 
of  the  Carnot  cycle  unfeasible  in  the  case  of  hot  air. 
Hence  recourse  has  been  had  to  certain  of  the  isodi- 
abatic  cycles  in  the  attempt  to  improve  the  work 
diagram. 

The  ideal  cycle  for  the  Stirling  hot-air  engine  con- 
sists of  the  following  events: 

(1)  Heating  at  constant  volume  by  passage  of  ah* 
through  regenerator. 

(2)  Expansion  at  constant  temperature  in  contact 
with  the  hot  surface  of  the  furnace. 

158 


HOT-AIR  ENGINES. 


159 


(3)  Cooling  at  constant  volume  by  return  through  the 
regenerator. 

(4)  Compression    at    constant    temperature    in    con- 
tact with  the  cooling  pipes. 

The  diagrams  for  such  a  cycle  are  shown  in  Fig.  60. 


FIG.  60. 

The  criterion  for  such  a  cycle  is  that  the  heat  re- 
jected at  any  temperature  T  along  be  shall  equal  that 
received  at  the  same  temperature  along  da.  Hence 


As  these  equations  both  refer  to  the  same  isothermal 
it  follows  that 

— •  =  -T2,     or    log  vt  +  log  ct  =  log  v2  +  log  c^ 


whence 


Or 


160        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


That  is,  the  lines  ad  and  be  are  "  isodiabatic/'  as 
they  satisfy  the  condition  that  the  ratio  of  the  volumes 
at  the  points  of  intersection  with  any  isothermal  is  a 
constant. 

The  ideal  cycle  of  the  Ericsson  engine  is  similar  to 
that  of  the  Stirling  except  that  the  heating  and  cooling 
occur  at  constant  pressure  instead  of  at  constant  volume. 


a     6 


FIG.  61. 


The  ideal  diagrams  for  such  a  cycle  are  shown  in 
Fig.  61.    In  this  case 


CpdT-(cp-cv) 
whence 


Pi 


Hence  these  curves  are   "  isodiabatic,"  since  the  ratio 
of  the  pressures  is  a  constant. 

Both  the  Stirling  and  the  Ericsson  cycle  give  well- 
shaped  indicator-cards  and  are  thus  better  than  the 
Carnot  cycle  mechanically. 


HOT-AIR  ENGINES. 


161 


General  Properties  of  Gas  Cycles. — It  was  shown  in 
Chapter  II  that  most  changes  of  condition  of  gaseous 
mixtures  can  be  represented  as  special  cases  of  the 
polytrope  pvn  =  const.,  where  the  specific  heat  of  the 

n  —  k 


change   is   defined   by  the   equation  c  =  cv 


We 


may  then  consider  the  general  gas  cycle  to  consist  of 
two  pairs  of  such  polytropic  curves,  whose  specific  heats 
are  c\  and  c2,  respectively,  where  c2>ci. 

Let    abed    (Fig.    62)    represent    such   a    cycle.    As 
4(j)abc=  J^>cda  it  follows  that 


or 


But  C2>ci,  and  therefore  loge 


TT, 


0.     There  thus 


exists  between  the  temperatures  at  the  four  corners  of 
the  cycle  .the  simple  relation, 


162        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Operating  as  a  motor  the  work  developed  is  equal 
to  the  difference  between  the  heat  received  and  the 
heat  rejected,  or 


Eliminating  T3  by  means  of  the  preceding  relation, 
this  reduces  to 


The  thermal  efficiency  is  therefore 


_ 
ri    T{cl(T-T2)+c2(f1-Tn' 

while    its    relative    efficiency    as    compared    with    the 
Carnot  cycle  is 


If  the  upper  and  lower  temperatures  are  maintained 
constant  it  is  evident  that  by  considering  point  a  to 
be  fixed  and  c  to  move  along  the  isothermal  T\,  T  can 
be  made  to  assume  all  intermediate  values.  During 
such  a  complete  change  the  external  work  would  pass 
from  zero  through  a  maximum  back  to  zero.  W  is 
therefore  a  function  of  T  and  the  value  of  T  which  will 

make  W  a  maximum  can  be  found  by  setting  -^  =  0. 


HOT-AIR  ENGINES. 


163 


Thus 


dAW 
dT  " 


or 


The  substitution  of  this  value  of  T  in  the  expression 
for  work  gives 


Under  these  conditions  the  heat  received  becomes 


and  the  thermal  efficiency 


The  relative  value  of  this  cycle  as  compared  with  the 
Carnot  is  therefore 


Carnot 

Special  Case.  —  If  c2  =  on    the  cycle  consists  of 
two  isothermals  and  two  poly  tropic  curves,  as  shown 


164       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

in  Fig.  63.  The  expression  for  the  maximum  amount 
of  work  (c2— Ci}(VT~\  —  \/T2)2  gives  an  infinite  result, 
because  no  limitation  has  been  placed  upon  the  isother- 
mal expansion.  Although  the  general  expressions 
possess  indeterminate  forms  a  special  solution  can  be 
obtained  directly  from  the  T</>-diagram. 


10  T,  Ci 


FIG.  63. 

The  work  developed  is  equal  to  that  developed  in  a 
Carnot  cycle, 


W 


Vb 


or 


PC 


The  thermal  efficiency  of  the  cyle  is  therefore 


HOT-AIR  ENGINES.  165 

and  its  relative  efficiency  as  compared  with  the  Carnot 
cycle  is 


PC 


PC 


(a)  When  c\  =  cv  this  reduces  to  the  Stirling  cycle, 
consisting  of  two  isothermals  and  two  constant-volume 
curves.  In  this  case  it  is  more  convenient  to  use  the 

ratio  of  volumes  —  instead  of  the  pressures  —  • 

Vb  PC 

(6)  When  c\  =  Cp  this  reduces  to  the  Ericsson  cycle, 
consisting  of  two  isothermals  and  two  constant-pressure 
curves. 

Of  course  the  above  discussion  applies  to  the  case 
when  a  regenerator  is  not  used. 

Other  special  cases  will  be  considered  in  the  Chapter 
on  Gas-engines. 

The  Non-regenerative  Stirling  and  Ericsson  Cycles.  — 
Inspection  of  the  formula  for  efficiency  or  of  Fig.  63 
shows  that  the  efficiency  increases  the  smaller  the 
relative  magnitude  of  the  heat  c\(T\  —  T2)  as  compared 
with  that  received  during  the  isothermal  process. 
That  is,  the  greater  the  range  of  pressure  in  the  Ericsson 
cycle  and  the  smaller  the  clearance  in  the  Stirling  cycle 
the  higher  the  efficiencies.  As  the  heat  received  during 
the  isothermal  change  increases  the  efficiency  of  each 
cycle  approaches  that  of  the  Carnot  as  a  limit. 


166      THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Example.  —  The  small  Ericsson  pumping-engine  (Stir- 
ling cycle  without  regenerator)  has  a  piston  8  ins.  in 
diameter  and  a  stroke  of  4  ins.  The  clearance  is  200 
per  cent,  of  the  piston  displacement.  Assuming  the 
temperature  of  the  fire  to  be  2500°  F.  abs.  and  that 
of  the  water-jacket  600°  F.,  find  the  theoretical  and 
relative  effic;encies. 


(2500-  600)  X 


0.169(2500-600)  + 

=  13.4  per  cent. 
2500-600 


Carnot  2500 

17.6  per  cent. 


76.0  per  cent. 


^Carnot 

The  Ericsson  Pumping-engine. — The  small  Ericsson 
pumping-engine  works  but  approximately  upon  the 
Sterling  cycle,  as  the  displacer  and  working  piston  are 
not  timed  so  to  operate  that  the  different  thermal  events 
are  entirely  separate.  This  action,  combined  with  the 
transference  of  heat  between  the  air  and  the  cylinder 
walls,  results  in  a  rounding  off  of  the  corners  of  the  ideal 
cycle,  giving  an  indicator  card  such  as  is  shown  in 


HOT-AIR  ENGINES.  167 

Fig  64.  It  is  to  be  further  noticed  that  heating  and 
cooling  at  constant  volume  exist  only  for  a  moment 
at  the  ends  of  the  stroke.  This  is  due  to  the  almost 
entire  absence  of  any  regenerator.  The  only  operation 
which  in  any  way  approximates  that  of  a  regenerator 
is  the  sudden  transference  of  the  air  from  one  end  of 
the  cylinder  to  the  other,  causing  it  to-  move  in  a  thin 
sheet  through  the  narrow  space  between  the  displacer 
and  the  cylinder.  It  thus  comes  into  contact  with 
metal  of  varying  temperatures,  and  is  thus  partly  heated 


FIG   64. 

or  cooled  during  transmission,  the  rest  of  the  heating 
or  cooling  occurs  during  the  operation  of  the  working 
piston. 

This  engine  in  itself  is  of  no  great  interest  to  the 
engineer,  but  the  T^-analysis  of  its  indicator  card  is  of 
value  in  that  it  permits  of  the  direct  application  of  the 
principles  already  discussed  in  the  Chapter  on  Perfect 
Gases  without  introducing  the  various  difficulties  to 
be  met  with  in  gas-engine  cards. 

The  same  charge  of  air  is  used  continuously  in  the 
Ericsson  engine,  the  heat  being  received  and  rejected 
through  the  walls  of  the  cylinder,  and  the  indicator  card 


168       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

thus  represents  a  closed  cycle  of  a  definite  mass  of  air. 
It  is  true  that  during  the  movement  of  the  displacer  the 
air  is  partly  in  contact  with  both  the  heated  and  jacketed 
portions  of  the  cyclinder  and  is  therefore  not  of  uniform 
temperature,  and  similarly  when  the  displacer  is  •  at 
rest,  although  most  of  the  air  is  in  contact  with  either 
the  source  or  the  refrigerator,  part  of  it  is  nevertheless 
in  the  narrow  space  outside  the  displacer  and  thus  at  a 
different  temperature  from  that  of  the  main  portion. 
There  is  thus  at  no  instant  a  uniform  temperature 
existing  throughout  the  entire  mass,  but  from  the  known 
pressure  and  volume  the  average  temperature  is  deter- 
minable. 

The  range  of  both  pressure  and  average  temperature 
is  so  small  that  the  air  follows  appreciably  the  laws  of  a 
perfect  gas,  so  that  cp  and  cv  are  constant. 

The  mass  of  air  is  unknown,  so  that  only  the  ratio  of 
the  specific  volumes  and  not  their  absolute  values  are 
known  for  different  positions  of  the  piston.  The  only 
item  definitely  known  is  therefore  the  pressure  as 
measured  from  the  indicator  card.  Therefore  in  the 
T^-projectioh  the  pressure  will  be  the  only  property 
definitely  known,  but  the  ratio  of  the  average  tempera- 
tures is  known  for  any  two  positions  of  the  piston. 
Furthermore  the  indicated  work  being  known  the  area  of 
the  indicator  card  in  the  T^-plane  represents  the  heat 
equivalent  of  this  work,  and  thus  the  B.T.U.  per  unit 
area  are  determinable.  Evidently  if  the  mean  tern- 


HOT-AIR  ENGINES.  169 

perature  should  be  obtained  experimentally  for  any 
position  of  the  piston,  the  weight  of  air  in  the  cylinder 
could  be  computed  and  thus  the  scales  of  temperature 
and  entropy  become  known. 

Fig.  65  shows  the  TV-  and  ^-projections  of  the 
pr-card  by  the  methods  outlined  in  Chapter  II.  Be- 
cause of  the  large  clearance  of  the  engine  it  was  possible 
to  economize  space  by  placing  the  T<£-projection  to  the 
left  of  the  TV-projection.  As  soon  as  one  becomes 
familiarized  with  the  methods  the  three  diagrams  can 
be  superimposed  without  detracting  any  from  their 
value. 

The  indicated  power  developed  per  cycle  is  obtained 
from  the  mean  effective  pressure  and  the  dimensions 
of  the  engine.  From  this,  together  with  the  areas  of 
the  two  diagrams,  the  foot-pounds  and  the  B.T.U.  per 
square  inch  may  be  determined  in  the  pv-  and  T<j>- 
projections  respectively. 

The  heat  received  and  rejected  by  the  air  per  stroke 
may  now  be  measured  as  the  surface  scale  of  the  T<f>- 
diagram  is  known,  although  the  scale  of  the  coordinates 
is  unknown. 

The  heat  theoretically  available  per  cycle  can  be 
found  in  any  given  case  by  measuring  the  fuel  and 
multiplying  the  weight  per  cycle  by  its  calorific  value. 
The  difference  between  the  heat  available  and  the  heat 
received  represents  the  heat  lost  up  the  flue  and  to  the 
surrounding  atmosphere,  cooling  water,  etc. 


170       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


As  a  basis  of  comparison  the  Stirling  cycle  between 
the  same  temperature  and  volume  limits  is  also  given. 


CHAPTER  IX. 

THE  TEMPERATURE-ENTROPY   DIAGRAM   APPLIED  TO 
GAS-ENGINE  CYCLES. 

No  attempt  will  be  made  here  to  trace  the  heat 
losses  due  to  radiation  and  the  cooling  water  in  the 
jackets,  but  the  cylinder  and  piston  will  be  considered 
impermeable  to  heat  in  all  cases.  Thus  by  a  compari- 
son of  the  ideal  cards  for  different  cycles  the  gain  due 
to  initial  compression  and  the  loss  from  incomplete 
expansion  may  be  more  clearly  defined. 

The  Lenoir,  Otto,  Atkinson,  and  Diesel  Cycles. — The 
Lenoir  cycle  was  introduced  in  1860.  Its  thermo- 
dynamic  principles  were  retained  in  the  different  free- 
piston  engines.  These  were  uneconomical  and  noisy 
and  have  disappeared.  The  only  remaining  example  is 
the  Bischoff,  a  simple  small  vertical  engine. 

The  Lenoir  cycle  consists  of  the  following  events: 

(1)  During  the  first  part  of  the  forward  stroke  a  fresh 
explosive  mixture  is  drawn  in  by  the  piston  (aA  in 
Fig.  66). 

(2)  A  little  before  half-stroke  is  reached  the  supply- 
valve  closes  and  the  explosion  occurs.     In  reality  the 

171 


172       THE  TEMPERATURE-EXTROPy  DIAGRAM. 

combustion  requires  an  appreciable  time  for  its  com- 
pletion, and  thus  the  heating  takes  place  while  the 
piston  is  moving  forward,  i.e.,  at  increasing  volume, 
but  for  the  ideal  case  the  explosion  will  be  considered 
instantaneous,  and  hence  the  heating  will  be  at  con- 
stant volume;  along  the  line  AB. 
(3)  The  rest  of  the  stroke  represents  adiabatic  expan- 


FIG.  66. 

sion  of  the  heated  gas  down  to  initial  pressure;  along 
BC. 

(4)  The  return  stroke,  during  which  the  products  of 
combustion  are  exhausted.  Thermodynamically  this 
is  equivalent  to  cooling  at  constant  pressure;  along  CA. 

About  the  time  Otto  and  Langen  were  experimenting 
with  the  free-piston  engine,  Beau  de  Rochas  described 
a  cycle  which  would  make  possible  the  economical 


GAS-ENGINE  CYCLES.  173 

running  of  a  gas-engine.  This  was  embodied  by  Dr.  Otto 
in  his  silent  engine  in  1876  and  has  thus  become  asso- 
ciated, although  wrongly,  with  his  name. 

The  "Otto"  cycle  consists  of  the  following  events: 

(1)  The  drawing  into   the   cylinder  at   atmospheric 
pressure  of  a  new  explosive  mixture  throughout  one 
complete  stroke  (a A  in  Fig.  66).     The  volume  of  the 
charge  is  MA  and  consists  of  the  burnt  products  in 
the  clearance  space  Ma  from  the  last  charge  plus  the 
fresh  charge. 

(2)  The  adiabatic  compression  of  this  charge  on  the 
return  stroke  of  the  piston  AD.     This  compression  of 
the  gas  into  the  clearance  space  is  done  at  the  expense 
of  the  energy  in  the  fly-wheel. 

(3)  The  ignition  and  explosion  of  the  charge  while 
the  piston  is  at  rest  at  the  dead-centre,  thus  increasing 
the  pressure  and  temperature  at  constant  volume ;  along 
DE.    Assuming  that  the  same  quantity  of  mixture  is 
used  by  both  the  Lenoir  and  Otto  engine,  the  heat 
generated  by  the  explosion  will  be  the  same  in  both 
cases,  i.e.,  the  areas  under  the  curves  AB  and  DE  are 
equal  in  the  T^-diagram. 

(4)  The  expansion  of  the  heated  gases  throughout 
the  entire  stroke,  assumed  adiabatic;  along  EF. 

(5)  The  drop  in  pressure  due  to  the  opening  of  the 
exhaust-valve  while  the  piston  is  at  the  end  of  the 
stroke.    This    is    equivalent    to    cooling    at    constant 
volume;  along  FA. 


174        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

(6)  The  exhaust  of  the  burnt  gases  during  the  return 
stroke  Aa.  Changes  of  location  are  not  recorded  in 
the  TV-diagram. 

In  the  Atkinson  engine,  now  no  longer  made,  the 
cycle  was  the  same  as  the  Otto  up  to  the  point  F,  and 
then,  instead  of  releasing  the  hot  gas,  the  expansion 
stroke  was  lengthened  by  means  of  an  ingenious 
mechanism  permitting  the  adiabatic  expansion  down 
to  back  pressure,  as  represented  by  FG.  Then  the 
exhaust  stroke  was  from  G  to  A,  which  thermo- 
dynamically  is  equivalent  to  cooling  at  constant 
pressure. 

Comparing  the  Atkinson  and  Otto  cycles  it  is  at 
once  evident  that  there  is  a  loss  of  work  and  of  heat 
equal  to  AFG  in  the  pv-  and  T<£-planes,  respectively, 
due  to  incomplete  expansion. 

A  comparison  of  the  Atkinson  and  Lenoir  cycles 
shows  that  as  the  heat  received  in  both  is  the  same 
while  that  rejected  by  the  Lenoir  engine  is  the  greater 
(compare  areas  under  CA  and  GA),  the  efficiency  of 
the  Atkinson  is  the  greater. 

Theoretically,  then,  the  Atkinson  engine  has  the  most 
perfect  cycle  of  the  three,  but  nevertheless  it  has  been 
entirely  superseded  by  the  Otto  engine.  The  reason 
for  this  becomes  at  once  apparent  from  the  diagram. 
Even  if  the  area  AFG,  rejected  by  the  Otto  engine, 
due  to  incomplete  expansion,  were  just  equal  to  the 
area  under  GC  of  the  Lenoir  exhaust  stroke,  the  Otto 


GAS-ENGINE  CYCLES.  175 

would  still  be  preferable  to  the  latter  because  the  same 
amount  of  power  could  be  developed  with  a  smaller 
engine.  Suppose,  now,  the  clearance  space  in  the 
above  Otto  engine  were  decreased,  so  that  the  adiabatic 
compression  would  heat  the  gas  to  a  higher  initial  tem- 
perature, as  AD'.  The  explosion  would  now  occur 
along  the  constant- volume  curve  D'E',  where  the  area 
under  D'E'  is  equal  to  that  under  DE,  as  the  same 
heat  is  generated  in  both  cases.  The  adiabatic  expan- 
sion would  now  be  down  E'F'  and  the  exhaust  would 
be  along  F'A.  Hence  the  same  Otto  engine  with 
increased  initial  compression  due  to  decreased  clearance 
would  .give  increased  efficiency,  as  the  heat  rejected 
under  F'A  is  less  than  that  rejected  under  FA.  This 
engine  would  now  be  better  than  the  Lenoir,  both 
mechanically  and  thermodynamically.  Furthermore 
the  loss  due  to  incomplete  expansion  becomes  less 
because  the  heat  thus  rejected  is  reduced  from  AFG 
to  AF'G'.  That  is,  the  higher  the  initial  compression 
the  less  the  theoretical  superiority  of  the  Atkinson  over 
the  Otto  engine.  And  in  the  actual  engine  the  increased 
complexity,  size,  friction  loss,  and  danger  of  the 
Atkinson  more  than  counterbalanced  the  theoretical 
superiority.  Hence  the  Otto  engine  is  practically 
the  most  efficient  of  the  three. 

In  the  Otto  cycle  the  temperature  at  the  end  of 
compression  is  not  very  high  relatively,  so  that  during 
the  first  part  of  the  combustion  the  working  fluid  is 


176        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

much  colder  than  the  source  of  heat  and  does  not 
attain  this  high  temperature  until  the  end  of  the  com- 
bustion is  reached.  The  heat  is  thus  received  at  con- 
stant volume  and  increasing  temperature  instead  of 
at  constant  temperature  and  increasing  volume.  It 
was  shown  in  the  first  chapter  that  any  such  deviation 
from  a  Carnot  cycle  means  a  drop  in  efficiency.  This 
led  Diesel  to  invent  a  cycle  during  which  most  of  the 
heat  should  be  received  at  the  highest  available  tem- 
perature. His  method  is  to  compress  the  air  initially 
up  to  about  five  hundred  pounds  pressure  to  the  square 
inch,  so  that  its  temperature  is  above  the  ignition- 
point  of  the  combustible  to  be  used.  The  injection 
of  a  small  quantity  of  fuel  causes  the  temperature 
to  increase  still  further  at  constant  volume  up  to  that 
of  the  combustion;  then  as  the  piston  moves  forward 
the  temperature  of  the  gas  is  maintained  nearly  constant 
by  the  injection  and  combustion  of  further  fuel.  This 
lasts  for  about  one-tenth  of  the  stroke.  The  indicator- 
cards  taken  from  such  a  motor  show  that  the  desired 
regulation  is  not  perfect,  the  temperature  sometimes 
rising,  sometimes  falling.  This,  however,  only  affects 
the  magnitude  of  the  gain  from  such  a  cycle,  as  all  of 
the  heat  generated  on  the  forward  stroke  is  trans- 
mitted to  the  working  fluid  at  an  efficiency  corre- 
sponding to  that  of  the  upper  part  of  the  Otto  cycle. 
The  cards  show  also  that  the  expansion  may  or  may 
not  be  carried  down  to  the  back  pressure. 


GAS-ENGLVE  CYCLES. 


177 


Fig.  07  shows  the  ideal  diagrams  for  the  Otto,  Atkin- 
son, and  Diesel  cycles  for  the  same  quantity  of  heat; 
that  is,  the  areas  under  be  and  b'gc'  are  the  same  in  the 
T^-plane.  The  change  from  b  to  br  shows  the  in- 
creased compression  in  the  Diesel  motor,  b'  being  at  or 
above  the  temperature  of  ignition.  The  heat  received 


g     c'     c 


FIG.  67. 

along  b'g  is  not  received  under  the  most  efficient  con- 
ditions, but  still  with  an  efficiency  equal  to  that  of  the 
best  part  of  the  Otto  cycle,  while  that  received  along 
the  "  isothermal  combustion  line  "  gc'  is  obtained  under 
conditions  of  maximum  efficiency.  The  effect,  as  is 
clearly  shown  by  the  diagram,  is  to  increase  the  amount 
of  heat  changed  into  work  and  to  dimmish  the  heat 
rejected.  On  the  return  stroke  the  conditions  of 
the  Carnot  cycle  are  more  closely  approximated  in 


178        THE    TEMPERATURE-ENTROPY  DIAGRAM. 


the  Atkinson  and  Diesel  cycles  where  the  heat  is  re- 
jected at  constant  pressure  than  in  the  Otto  cycle 
where  it  is  rejected  at  constant  volume,  as  lines  of 
constant  pressure  deviate  from  isothermals  less  than 
do  lines  of  constant  volume. 

The  Brayton  or  Joule  Cycle. — Up  to  the  present  time 
the  Otto  cycle  has  been  almost  exclusively  employed 


FIG,  68. 

because  of  its  ease  of  application  in  the  reciprocating 
type  of  motor,  but  occasional  attempts  have  also  been 
made  to  heat  the  fuel  under  constant  pressure.  Such 
a  cycle  is  now  assuming  importance  as  offering  the 
most  probable  solution  of  the  gas-turbine  problem. 
Proposed  by  Joule  this  cycle  is  perhaps  better  known 
through  its  application  in  the  Brayton  engine.  It 
consists  of  the  following  events  (Fig.  68); 

(1)  The  charging  stroke  of  the  compressor  cylinder, 
ab. 

(2)  Adiabatic  compression  in  the  compressor,  be. 


GAS-ENGINE  CYCLES,  179 

(3)  Discharge  from  the  compressor  at  constant  pres- 
sure, cd. 

(4)  During  the  transfer  from  the  pump  to  the  working 
cylinder  (or  to  the  nozzle  of  the  turbine)  the  charge 
receives    heat    either   from    external   sources   or   from 
combustion  under  constant  pressure,  but  with  increasing 
volume  and  temperature. 

(5)  Admission  stroke  of  the  working  cylinder,  de. 
Processes  (3),  (4),  and  (5)  occur  simultaneously.. 

(6)  Adiabatic  expansion  in  the  working  cylinder  or 
in  the  nozzle,  ef. 

(7)  Discharge  from  the  working  cylinder  against  a 
constant  back  pressure,  fa. 

The  net  result  is  thus  represented  by  the  cycle  beef, 
consisting  of  two  isentropic  and  two  constant-pressure 
curves. 

It  is  at  once  evident  that  the  efficiency  of  the  Brayton 
cycle,  like  that  of  the  Otto,  increases  with  increased 
initial  compression. 

It  is  instructive  to  compare  the  three  ideal  cycles  of 
Carnot,  Otto,  and  Brayton — involving  as  they  do  heating 
at  constant  temperature,  constant  volume,  and  constant 
pressure — under  different  conditions. 

Let  abed,  abc'd',  and  abc"d"  (Fig.  69)  represent  the 
Otto,  Brayton,  and  Carnot  cycles  respectively,  for  the 
same  weight  of  charge  working  with  the  same  initial 
compression.  The  areas  under  be,  be',  and  be"  will 
then  be  equal. 


180       THE  TEMPERATURi'.-hNTROPY  DIAGRAM. 

The  thermal  efficiencies  of  the  cycles  are  found 
follows  : 


cv(Tc-Tb) 


_ 


Td-Tg 

Tc-Tb 


=  I  -  Tfr,    or    1  -  (  — 

Tb'  \Va 


k-l 


^Brayton  — 


that  is, 


Tj-Tg 

ZV-n 


Ta 

•*•     b 

^Otto 


^Carnof 


FIG.  69. 


GAS-ENGINE  CYCLES.  181 

Such  a  case  might  arise  when  all  three  engines  were 
using  the  same  combustible,  thus  necessitating  that 
the  initial  pressure  be  limited  to  prevent  pre-ignition. 
Since  the  efficiency  of  all  three  cycles  will  be  the  same, 
a  decision  as  to  the  most  advantageous  to  use  must  be 
based  upon  other  considerations. 

The  differences  between  the  three  cycles  are  best 
shown  in  tabular  form : 

Pressure  Range.  Volume  Range,  Temp.  Range. 

Brayton Minimum  Intermediate  Intermediate 

Otto Maximum  Minimum  Maximum 

Carnot Intermediate  Maximum  Minimum 

From  the  table  the  Carnot  is  shown  to  be  the  least 
desirable  in  that  it  requires  the  largest  engine,  although 
the  fluctuations  in  its  rotative  effect  are  not  as  great 
as  in  the  Otto,  so  that  the  working  parts  would  not 
need  to  be  as  heavy  as  in  the  latter.  But  besides  this 
the  Carnot  cycle  has  to  be  discarded  as  unfeasible 
because  although  the  Diesel  motor  approximates  it  on 
the  forward  stroke  no  gas-engine  has  as  yet  been  in- 
vented to  give  isothermal  compression. 

The  Brayton  cycle  has  the  minimum  change  of  pres- 
sure and  therefore  the  least  fluctuating  rotative  effect. 
Its  volume  range  is,  however,  greater  than  that  of  the 
Otto.  But  again  the  maximum  temperature  reached 
in  the  Brayton  is  less  than  in  the  Otto,  so  that  less  jacket 
cooling  of  the  cylinder  will  be  necessary  and  the  inter- 
change of  heat  between  gas-metal-gas  will  also  be  less. 


182       THE    TEMPERATURE-ENTROPY  DIAGRAM. 


By  keeping  the  gas  and  air  separate,  as  in  the  Diesel 
motor,  it  is  possible  to  utilize  much  higher  initial  com- 
pression, and  then  the  only  limit  theoretically  is  the 
resistance  of  the  machine  to  pressure  or  temperature  or 
both.  We  will  next  compare  the  Brayton  and  Otto 
cycles  with  reference  to  maximum  pressure  and  maxi- 
mum temperature. 
Let  bcc'  (Fig.  70)  represent  the  maximum  safe  pressure, 


FIG.  70. 

whether  it  be  the  sustained  pressure  of  the  Brayton 
cycle  or  the  momentary  maximum  of  the  Otto.  Fur- 
ther, let  ad  be  atmospheric  pressure,  the  lowest  pressure 
of  exhaust. 

In  the  Brayton  cycle  initial  compression  is  carried 
up  to  6,  while  in  the  Otto  cycle  it  must  stop  at  some 
intermediate  point  6',  such  that  the  further  increase 


GAS-ENGINE  CYCLES. 


183 


due  to  heating  will  carry  it  just  to  c'.  As  the  charges 
are  identical  the  area  under  be  equals  the  area  under 
b'c'.  But  the  heat  rejected  in  the  Otto  exceeds  that  in 
the  Brayton.  Whence  it  follows  that  given  a  Brayton 
and  an  Otto  engine  working  between  the  same  limits  of 
pressure  the  former  is  the  more  efficient  and  possesses 
the  smaller  temperature  range. 


d 


FIG.  71. 

Similarly  let  ccf  (Fig.  71)  represent  the  safe  maximum 
temperature.  Then  abed  and  dh'c'd'  will  represent  the 
Brayton  and  Otto  cycles,  respectively,  working  between 
atmospheric  pressure  and  temperature  and  maximum 
temperature  cc'.  Again  the  Brayton  cycle  shows  the 
greater  efficiency,  although  both  possess  the  same 
temperature  range,  and  the  Otto  cycle  has  a  trifle  the 
smaller  pressure  range. 


184        THE   TEMPERATURE-ENTROPY   DIAGRAM. 

Deduction  from  the  General  Theory  of  Gas  Cycles. — 
On  pages  101-163  we  discussed  the  general  case  of  a  gas 
cycle  consisting  of  two  pairs  of  polytropic  curves,  and 


FIG.  72. 

obtained  as  the  condition  of  maximum  work  per  cycle 
that 


This  gave  for  maximum  work  and  the  corresponding 
efficiency  the  expressions, 


and 


Second  Special  Case. — If  Ci  =  0  the  cycle  consists  of 
two  isentropic  and  two  polytropic  curves,  as  shown  in 


GAS-ENGINE  CYCLES.  185 

Fig.  72.     The  maximum  work  per  cycle  between  given 
temperature  limits  reduces  therefore  to 


and  the  corresponding  thermal  efficiency  to 

_  VT\  -  \/T~2    VT^Tl  -T2    T-T2 

>5max.Work=  ^  VT^  ~T~ 

whence  as  on  p.  176, 


The  dimensions  of  the  engine  to  give  the  maximum 
amount  of  work  per  cycle  between  any  given  tempera- 
ture limits  may  be  found  as  follows  : 

The  equation  of  the  TV-projection  of  an  adiabatic 
gives 


whence 


v\k-*^  / Clearance  N*"1 

~~  \Clearance  +  piston  displacement/ 

m  m  \rn 


or 

P.D.(^r 

Clearance  =  — 


1- 


186        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Variation  of  Efficiency  with  Load. — The  efficiency  of 
the  internal-combustion  engine  varies  not  only  with 
the  type  of  cycle  utilized,  but  also  with  the  form  of 
regulation  adopted  in  any  particular  engine. 

A.  Engines  working  upon  the  Otto  cycle  are  governed 
by  the  following  methods: 

I.  The  omission  of  an  occasional  charge,  known  as 

hit-or-miss  regulation. 

II.  Throttling  of  the  fuel  supply,  known  as  quality 
regulation. 

III.  Variation  of  the  amount  of  charge,  obtained  by 

throttling  the  mixture  or  by  a  variable  cut- 
off, known  as  quantity  regulation. 

IV.  Regulation  by  varying  the  time  of  ignition. 

V.  Various  compound  methods  involving  two  or 
more  of  the  above. 

The  effects  of  these  various  methods  upon  the  efficiency 
of  the  engine  under  variable  load  can  be  clearly  illus- 
trated with  the  ideal  cycle  in  the  T(f>  plane. 

I.  Hit-or-miss  Regulation. — Whenever  a  charge  of 
fuel  is  omitted  the  scavenging  action  of  the  air  com- 
bined with  the  constant  action  of  the  jackets  serves  to 
cool  the  cylinder  walls  and  the  admission  ports  to  a 
temperature  below  the  average,  so  that  the  next  charge 
will  be  cooler  and  of  greater  density. 

Therefore  the  first  explosion  immediately  following 
a  "  miss  "  stroke  will  be  more  powerful  than  the  average 
as  it  contains  a  greater  quantity  of  fuel.  Thus  if  abc'd', 


GAS-ENGINE  CYCLES. 


187 


vig.  73,  represent  the  average  cycle,  the  cycle  following 
the  "  miss  "  will  be  represented  by  some  diagram  such 
as  ahc"&" .  In  case  of  light  load  it  may  happen  that 
.two  or  more  miss  strokes  occur  successively,  in  which 
case  the  walls  may  be  cooled  to  such  an  extent  that  the 
resulting  low  temperature  of  the  next  charge  retards 
its  combustion.  This  is  equivalent  to  exhausting  part 


FIG.  73. 

of  the  charge  unburned  and  results  in  a  smaller  heat 
diagram  abed. 

It  will  be  noticed  that  with  constant  port  openings 
and  constant  speed  the  throttling  action  during  admission 
is  always  the  same,  so  that  the  pressure  at  the  end  of 
admission  is  always  practically  the  same.  Hence  if 
the  minor  variations  in  percentage  mixture  caused  by 
the  varying  density  and  composition  of  the  clearance 
gases  may  be  overlooked,  the  effect  of  a  miss  stroke  is 
simply  to  vary  the  power  of  the  succeeding  stroke  but 
not  its  efficiency. 


188        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Variation  in  load  is  met  automatically  by  the  engine 
by  varying  the  frequency  of  the  miss  strokes.  There- 
fore this  type  of  regulation  must  give  practically  the 
same  thermal  efficiency  for  all  loads,  but  it  has  the 
mechanical  disadvantage  of  .requiring  more  massive 
construction  and  heavier  fly-wheels  if  good  regulation 
in  speed  is  to  be  obtained.  In  practice  engines  governed 
by  hit-or-miss  regulation  usually  show  a  somewhat 
greater  fuel  economy  than  is  obtained  by  the  other 
methods. 

II.  Quality  Regulation. — A  full  charge  of  fuel  and  air 
is  taken  each  suction  stroke,  but  the  percentage  mixture 
is  varied  by  the  governor  which  controls  a  throttle 
valve  placed  in  the  fuel  supply  pipe.  The  initial  pres- 
sure being  the  same  the  efficiency  should  apparently 
be  the  same,  but  experiments  made  upon  explosive 
mixtures*  of  fuel  and  air  have  shown  that  the  rate 
of  combustion  is  retarded  by  dilution,  and  this  dilution 
may  readily  be  so  great  as  to  prevent  combustion  or  at 
least  to  delay  it  until  after  the  charge  has  left  the 
cylinder.  This  unburned  or  wasted  heat,  combined 
with  the  greater  relative  effect  of  the  cylinder  walls 
upon  a  smaller  heat  charge  (see  next  chapter),  serves 
to  diminish  the  efficiency  as  the  load  falls  off. 

Thus  if  abed,  Fig.  74,  represent  the  heat  available 
at  full  load,  abc'd'  would  represent  the  heat  available 

*  See  report  by  Author  in  Technology  Quarterly,  Sept.,  1900,  pp. 
248-259. 


GAS-ENGINE  CYCLES. 


189 


for  some  smaller  load  if  it  were  not  for  the  above  men- 
tioned losses.  Due  to  dilution,  however,  a  certain  por- 
tion of  this  theoretically  available  heat,  say  c"c'd'd", 
either  is  lost  through  imperfect  combustion  or  is  so 
much  delayed  as  to  be  generated  after  the  charge  is 


FIG.  74. 


exhausted  and  must  therefore  be  added  to-  the  exhaust 
heat,  as  d'efg. 

III.  Quantity  Governing. — In  this  system  of  regula- 
tion the  percentage  mixture  is  adjusted  by  hand  to  give 
the  maximum  explosive  value  and  is  then  maintained 
constant,  but  the  quantity  of  the  mixture  admitted 
to  the  cylinder  is  controlled  by  a  valve  placed  in  the 
pipe  leading  .from  the  mixing  chamber  to  the  cylinder. 
The  action  of  the  governor  being  either  to  throttle  the 
valve  throughout  the  suction  stroke  or  to  close  it 
completely  at  varying  percents  of  the  stroke  so  as  to 
adjust  the  quantity  of  charge  to  the  load. 


190        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

a.  Throttling. — Regulation  by  throttling  involves  a 
drop  of  pressure  in  the  throttle  valve  so  that  the  pressure 
of  admission  is  less  than  the  pressure  of  discharge. 
This  produces  a  negative  loop  in  the  indicator  card 


FIG.  75. 

whose  area  measures  the  work  required  to  overcome  the 
friction  losses  during  admission  and  exhaust.  See 
Fig.  75. 

b.  Automatic  Cut-off.— Here  the  valve  is  kept  wide 
open  until  cut-off  occurs.     Hence  the  charge  is  admitted 


FIG.  76. 

at  atmospheric  pressure  up  to  cut-off,  and  from  that 
point  expands  adiabatically  to  the  end  of  the  stroke. 
On  the  compression  stroke  this  adiabatic  operation 
is  traversed  in  the  reverse  direction  until  atmospheric 
pressure  is  reached.  The  friction  loss  of  the  preceding 
method  is  therefore  avoided.  Figs.  76  and  77. 

The  weight  of  the  burnt  gases  is  practically  always  the 
same,  as  they  are  discharged  under  the  same  pressure, 


GAS-ENGINE   CYCLES. 


191 


while  the  weight  of  the  new  charge  varies  with  the 
amount  of  throttling  and  with  the  time  of  cut-off,  so 
that  the  proportion  between  the  burnt  gases  and  the 
new  charge  is  a  variable,  that  is,  the  burnt  gases  serve 
to  increase  the  dilution  of  the  gaseous  mixture  as  the 
load  decreases.  Quantity  regulation  thus  possesses  not 


FIG.  77. 

only  the  losses  due  to  its  throttling  action  but  also  to 
a  small  extent  those  due  to  quality  regulation. 

IV.  Regulation  by  Varying  the  Time  of  Ignition. — 
Ignition  must  occur  before  the  end  of  the  compression 
stroke  so  that  by  the  time  compression  is  finished  the 
flame  shall  have  had  opportunity  to  spread  throughout 
the  entire  charge  and  produce  maximum  pressure. 
For  any  given  load  the  spark  should  therefore  be 
adjusted  by  hand  so  as  to  give  best  results.  In  quality 
regulation  and  to  a  lesser  extent  in  quantity  regulation 
the  rate  of  flame  propagation  falls  off  as  the  load  de- 
creases, so  that  if  best  results  are  to  be  attained  the 
spark  must  be  correspondingly  advanced. 


192       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Advantage  is  taken  of  this  in  many  smaller  motors, 
as  for  example  in  automobiles  and  motor  boats,  to 
adjust  the  power  of  the  engine  to  any  desired  condition 
by  shifting  the  spark  by  hand.  Thus  practically  the 
same  quantity  of  fuel  is  admitted  each  stroke  and  more 
and  more  of  it  wasted  as  the  load  falls  off. 

Assuming  for  the  moment  that  the  explosion  is 
absolutely  instantaneous  the  effect  of  retarding  the 


FIG.  78. 


spark  is  clearly  shown  in  Fig.  78.  Let  abed  represent 
normal  running  under  full  load.  If  the  load  falls  off 
somewhat  the  spark  is  retarded  until  the  charge  has 
expanded  part  way  down  the  compression  line  ba, 
say  to  &';  it  then  burns  at  constant  volume  b'c'  gener- 
ating the  same  amount  of  heat  as  at  be  but  with  lower 
temperatures  and  hence  greater  entropy  increase. 
The  result  is  therefore  an  increase  in  the  exhaust  heat 


GAS-E.\GINE    CYCLES. 


193 


by  an  amount  dd'fg.  The  later  the  spark  the  greater 
the  increased  exhaust.  In  other  words  the  effect  is 
similar  to  that  which  would  be  obtained  by  an  in- 
creasing clearance  space. 

B.  Engines  working  upon  the  Diesel  cycle  are  governed 
by  varying  the  time  interval  during  which  fuel  is  admit- 
ted to  the  cylinder. 

Let  it  be  assumed  that  there  is  always  sufficient 
oxygen  present  to  insure  complete  combustion  of  the 


FIG.  79. 

fuel  at  maximum  load.  If  the  fuel  is  injected  so  as 
to  supply  heat  isothermally  the  shape  of  the  cycle 
with  increasing  load  is  shown  by  abcdef,  abcd'e'f,  etc., 
Fig.  79,  where  the  areas  under  bed,  bed',  etc.,  rep- 
resent the  heat  generated  by  the  fuel  and  de,  d'e',  etc., 
represent  the  adiabatic  expansion  from  the  moment 
the  fuel  supply  is  discontinued  to  the  end  of  the  stroke. 
It  is  evident  that  the  drop  of  temperature  de,  d'e', 


194        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

etc.,  experienced  by  the  additional  fuel  decreases  so 
that  the  average  efficiency  falls  off  as  the  load  increase?. 
If,  for  example,  injection  of  fuel  could  be  continued  to 
the  end  of  the  stroke,  the  ideal  efficiency  would  be 
but  slightly  greater  than  one  half  that  of  the  Carnot 
efficiency  although  at  ordinary  load  it  is  nearly  equal 
to  it. 

Comparison  of  the  Different  Gas-cycles.  —  Because 
of  its  successful  application  in  commercial  use  it  is 
necessary  to  deduce  an  analytical  expression  for  the 
efficiency  of  the  Diesel  cycle.  Certain  assumptions 
must  be  made  at  the  start  in  order  to  give  this 
cycle  that  simplicity  of  form  which  will  make  the 
analytical  expression  useable.  .  On  actual  indicator 
cards  taken  from  Diesel  engines  the  process  most  vari- 
able in  character  is  that  of  the  admission  of  the  fuel. 
This  fluctuates  more  or  less  in  magnitude  upon  the 
different  cards,  but  in  general  may  be  fairly  well  repre- 
sented by  a  constant  pressure  line.  Such  an  assumption 
is  justifiable  from  a  theoretical  point  of  view,  as  the 
Diesel  cycle  attempts  to  utilize  the  great  increase  in 
efficiency  which  comes  from  high  initial  compression. 
In  order  that  this  gain  may  be  a  maximum,  such  com- 
pression should  be  carried  to  the  upper  pressure  limit, 
so  that  when  the  fuel  is  injected  no  further  increase 
in  pressure  is  permissible  and  any  decrease  in  pressure 
would  diminish  the  temperature  range  of  the  heat. 
This  differs  from  the  original  Diesel  cycle  which  assumed 


GAS-ENGINE  CYCLES.  195 

isothermal  combustion,  and  was  based  upon  the  mis- 
taken idea  that  isothermal  generation  of  heat  was 
essential  to  the  attainment  of  maximum  efficiency. 
Assuming,  then,  that  the  ideal  Diesel  cycle  may  be 
represented  by  constant  pressure  reception  of  heat  and 
constant  volume  rejection  of  heat,  these  two  processes, 
being  connected  by  adiabatic  expansion  and  com- 
pression, it  follows  that  the  expression  of  thermal 
efficiency  is  equal  to  Cp  times  the  increase  in  temper- 
ature during  the  injection  of  the  fuel,  minus  Cv  times 
the  decrease  in  temperature  during  release,  divided  by 
the  heat  received.  A  fundamental  distinction  between 
this  expression  and  the  common  expressions  for  the 
Otto,  Joule,  and  Carnot  cycles  is  to  be  noted  in  the 
retention  of  the  specific  heats.  Thus  in  the  Otto  cycle 
Cv  appears  in  all  three  terms,  and  similarly  in  the 
Joule  cycle  Cp  appears  in  all  three  terms,  so  that  this 
common  factor  cancels  in  all  these  expressions  making 
the  efficiency  absolutely  independent  of  .the  character 
of  the  working  substance,  whereas  in  the  expression 
for  the  efficiency  of  the  Diesel  cycle  the  ratio  of  these 
specific  heats  remains,  so  that  the  efficiency  is  dependent 
to  a  certain  extent  upon  the  character  of  the  working 
fluid  used.  By  means  of  thermodynamic  relations 
easily  established  between  the  temperatures  and  the 
volumes  of  the  corners  of  the  cycle  it  is  possible  to  elim- 
inate from  the  expression  of  the  efficiency  all  the  tem- 
peratures and  to  retain  in  their  places  certain  ratios 


196        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

of  volumes.    The  expression  may  be  given  in  the  follow- 
ing final  form: 

/  Vol.  at  cut-off  \* 

\Vol.  of  clearance/  /Vol. of  clearance"1 

Mt.  =  l  — 


or 


/  Vol.  at  cut-off       ~\  I  Vol.  of  cl.  +  pis- 1 
AVol.of  clearance"   A       ton  dis.       / 

(s&Y-i 

Fff      i      ^  CL    '  (       CL        V"1 

-,/c.o.  1v\ci.+p.pJ 

H-cr-1) 

This  expression  is  of  the  same  form  as  that  of  the 
efficiency  of  the  Otto,  Joule,  and  Diesel  cycles,  except 
that  a  certain  coefficient  different  from  unity  is  intro- 
duced as  a  multiplier  in  the  last  term.  Thus,  as  we 
have  seen,  the  expression  for  the  Otto  efficiency  takes 

the  form : 

/       Q        y-i 

-VCL+RPJ 
An  investigation  of  this  coefficient : 


Cl. 


shows  us  that  as  the  cut-off  increases  the  numerator 
increases  more  rapidly  than  the  denominator,  since 
the  expression : 

/c.o.y 
V~cT/ ' 


GAS-ENGINE  CYCLES.  197 

as  k  is  greater  than  one,  increases  more  rapidly  than 
the  simple  ratio 


/C.0.\ 

rci/- 


Therefore  as  the  load  increases  the  value  of  this  coeffi- 
cient increases  so  that  the  efficiency  of  the  cycle  di- 
minishes. The  maximum  values  of  the  efficiency  of  the 
Diesel  cycle  are  therefore  obtained  with  the  smaller 
loads,  and  as  the  load  continues  to  decrease  this  value 
increases  and  approaches  as  a  limit  that  of  the  Otto 
cycle  which  theoretically  will  be  attained  at  zero  load. 
It  is  evident,  therefore,  that  with  the  same  com- 
pression the  Diesel  cycle  is  not  as  efficient  as  either 
the  Otto,  Joule,  or  Carnot  cycles,  but  that  its  great 
practical  efficiency  has  been  attained  by  making  use 
of  the  increase  in  efficiency  in  any  of  these  cycles  which 
may  be  produced  by  utilizing  smaller  clearance  space. 
This,  as  we  have  seen,  is  accomplished  by  keeping 
the  fuel  separate  from  the  air  during  the  compression 
and  injecting  the  fuel  whenever  the  desired  pressure 
has  been  attained.  As  soon  as  this  method  of  separate 
compression  is  introduced  there  is  no  longer  any  neces- 
sity of  utilizing  such  low  compressions  in  the  Otto  and 
Joule  cycles,  so  that  it  would  be  possible  to  have  in 
these  cycles  exactly  the  same  compression  as  that 
realized  in  the  Diesel  engine,  and  in  such  a  case  this 
cycle  would  be  inferior  to  either  of  the  others.  It  is 
probable  that,  as  advantage  is  being  taken  of  this 


198        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

high  compression,  the  initial  compression  would  be 
carried  to  the  ultimate  upper  limit  consistent  with 
safety,  so  that  no  further  increase  in  pressure  would  be 
permitted  during  the  burning  of  the  fuel.  In  such 
a  case  it  would  no  longer  be  possible  to  utilize  the  Otto 
cycle,  and  one  would  be  forced  back  upon  the  Joule 
cycle. 

In  case  compression  is  not  carried  to  the  maximum 
upper  limit  and  a  still  further  increase  is  permissible 
in  the  cylinder  during  combustion,  it  would  then  be 
possible  to  utilize  burning  at  constant  volume  and 
therefore  receive  the  heat  in  the  manner  adopted  in  the 
Otto  cycle.  If  in  connection  with  this  constant  volume 
reception  of  heat  there  could  at  the  same  time  be  com- 
bined a  constant  pressure  rejection  of  heat,  an  efficiency 
could  be  obtained  greater  than  that  of  either  the  Otto 
or  the  Joule  cycle,  A  cycle  thus  constituted  would 
evidently  be  but  the  old  Atkinson  cycle.  Proceeding 
in  a  manner  similar  to  that  adopted  in  the  case  of  the 
Diesel  cycle,  an  expression  for  the  efficiency  of  the 
Atkinson  cycle  can  be  deduced  in  terms  of  k  and  certain 
volume  ratios.  This  expression  may  be  given  the 
final  form: 

/Vol.  after  expansion      \    ,  Vol.  of    X*"1 
\Vol.  after  admission       /  /  clearance   \ 
~  /Vol.  after  expansion^*  """'  I   Vol.  after    I 
VVol.  after  admission/  "       \admission  / 

This  expression  is  seen  to  have  the  same  form  as  that 


GAS-ENGINE  CYCLES.  199 

of  the  Otto  cycle  with  the  exception  that  the  coefficient 
of  the  second  term  is  other  than  unity.  An  inspection 
of  this  coefficient: 


k 


/Vol.  after  expansion       \ 
\Vol.  after  admission       / 


Vol.  after  expansion\  k 
Vol.  after  admission/ 


shows  that  the  efficiency  is  dependent  upon  the  value 
of  k  for  the  given  substance  and  also  dependent  upon 
the  volume  after  expansion;  this  volume  necessarily 
increasing  as  the  quantity  of  charge  increases. 

As  the  volume  after  suction  may  be  considered  a 
fixed  quantity,  it  is  evident  that  as  the  volume  after 
expansion  increases  the  ratio  of  these  two  volumes 
increases  at  a  slower  rate  than  the  same  ratio  raised  to 
the  k  power,  so  that  with  increasing  load  the  denom- 
inator of  this  coefficient  increases  at  a  higher  rate  than 
the  numerator,  so  that  the  coefficient  as  a  whole 
decreases  in  value  as  the  load  increases,  a  result  which 
is  exactly  the  opposite  of  the  history  of  the  correspond- 
ing coefficient  for  the  efficiency  of  the  Diesel  cycle. 

An  inspection  of  these  two  cycles  in  the  temperature 
entropy  plane  makes  clear  this  fundamental  differ- 
ence in  these  two  cycles.  Start  with  the  same  initial 
compression  in  both  cycles.  In  the  Diesel  cycle,  on  the 
one  hand,  the  constant  pressure  curve  drawn  through 
the  end  of  compression  and  the  constant  volume  curve 


200      THE   TEMPERATURE-ENTROPY  DIAGRAM. 

drawn  through  the  beginning  of  compression  approach 
indefinitely  as  the  load  increases,  so  that  whereas  at 
the  beginning  or  at  a  very  light  load  the  efficiency 
is  equal  to  that  of  a  Carnot  cycle,  we  find  that  the  suc- 
cessive quantities  of  heat  taken  in  as  the  load  increases 
suffer  a  smaller  drop  in  temperature  than  the  earlier 
quantities,  and  therefore  are  utilized  less  efficiently, 
so  that  the  net  result  of  an  increasing  load  is  to  bring 
down  the  average  efficiency  at  which  the  heat  is  utilized. 
In  the  Atkinson  cycle,  on  the  other  hand,  since  a  con- 
stant volume  curve  drawn  through  the  end  of  compres- 
sion and  a  constant  pressure  curve  drawn  through  the 
beginning  of  compression  diverge  as  the  load  increases, 
the  range  of  temperature  experienced  by  successive 
quantities  of  heat  increases  so  that  the  efficiencies 
of  the  final  portions  are  greater  than  the  efficiencies 
of  the  earlier  portions,  and  therefore  the  result  of  an 
increasing  load  is  to  increase  the  average  efficiency 
at  which  the  heat  is  utilized.  As  the  load  falls  off 
this  coefficient  increases  in  value  and  approaches  unity 
as  a -limit,  this  value,  however,  being  attained  only  at 
zero  load,  when  the  efficiency  of  the  Atkinson  cycle 
will  be  reduced  to  that  of  the  Otto  cycle. 

This  discussion  makes  it  evident  that  the  Carnot 
cycle  is  not  the  most  efficient  cycle  when  adapted  to 
gas  engine  work.  The  requirement  that  all  the  heat 
should  be  received  at  the  highest  possible  temperature, 
and  that  all  the  heat  which  must  be  rejected  should 


OAS-EXGIXE  CYCLES.  201 

be  rejected  at  the  lowest  possible  temperature  in  order 
to  obtain  the  maximum  amount  of  work  from  a  given 
quantity  of  heat,  is  necessarily  fundamentally  true.  If 
the  source  of  heat  were  at  a  definite  upper  temperature, 
the  fulfillment  of  this  fundamental  requirement  would 
necessarily  require  the  use  of  the  Carnot  cycle,  but 
if  the  heat  is  generated  by  combustion  in  a  confined 
space  the  very  act  of  combustion  will  result  in  an  increase 
in  the  temperature  of  the  substance  in  this  confined 
space,  so  that  any  further  combustion  will  occur  at  a 
higher  temperature.  In  other  words,  successive  quan- 
tities of  heat  will  be  generated  in  the  confined  space 
at  successively  higher  temperatures.  In  fact,  to  prevent 
the  temperature  from  rising  in  order  to  obtain  the 
requirement  of  the  Carnot  cycle,  would  necessitate 
careful  adjustment  between  the  rate  of  fuel  consump- 
tion and  the  rate  of  work  production  by  the  piston. 

In  the  Diesel  cycle  the  heat  is  generated  at  such 
a  rate  that  the  pressure  remains  practically  constant. 
In  the  Otto  cycle  the  generation  of  heat  is  so  nearly 
instantaneous  that  the  process  occurs  practically  at 
constant  volume.  In  both  of  these  cases  the  temperature 
rises  continuously  throughout  the  period  of  com- 
bustion, and,  other  things  being  equal,  the  heat  ^received 
along  the  constant  pressure  curve  is  capable  of  better 
utilization  than  that  received  along  the  isothermal 
curve,  and  similarly  the  heat  received  at  constant 
volume  is  capable  of  better  utilization  than  that  received 


202       THE   TEMPERATURE-ENTROPY  DIAGRAM. 

at  constant  pressure.  This  is  very  clearly  illustrated 
by  the  relative  slopes  of  the  three  curves  in  the  tem- 
perature entropy  plane;  as  the  steeper  the  curve  the 
smaller  the  increase  in  entropy,  and  therefore  the 
smaller  the  amount  of  heat  which  must  eventually  be 
rejected.  This  analysis  shows,  therefore,  that  for 
the  same  initial  compression  the  constant  volume  curve 
is  the  one  which  has  the  highest  theoretical  possibili- 
ties. It  might  seem  at  first  sight  that  a  curve  even 
steeper  than  the  constant  volume  could  be  utilized. 
A  little  thought,  however,  will  show  that  such  a  process 
could  only  be  taking  place  while  the  piston  was  on  its 
return  stroke.  In  such  a  case,  therefore,  it  would 
be  better  to  wait  until  this  return  stroke  were  com- 
pletely finished,  so  that  all  of  the  heat  might  be  gener- 
ated at  the  higher  temperatures  consequent  upon  such 
an  increased  compression.  As  we  have  seen,  a  cycle 
consisting  of  constant  volume  production  of  heat,  a 
constant  pressure  rejection  of  heat,  combined  with 
adiabatic  compression  and  expansion,  gives  a  higher 
efficiency  than  the  straight  Otto  cycle.  Such  an  Atkin- 
son cycle,  however,  although  it  fulfills  the  requirements 
for  the  reception  of  heat  at  maximum  temperature, 
does  not  at  the  same  time  fulfill  the  requirements  for 
the  rejection  of  heat  at  minimum  temperature.  It 
is  true  that  the  Atkinson  cycle  expanding  down  to 
initial  pressure  probably  rejects  the  heat  at  as  low  a 
temperature  as  is  practicable,  but  there  nevertheless 


GAS-ENGINE  CYCLES.  203 

exists  the  theoretical  possibility  of  continuing  this 
expansion  down  to  the  initial  temperature  and  then 
closing  the  cycle  by  means  of  an  isothermal  com- 
pression. Possibly  such  an  operation  could  be  realized 
to  a  certain  extent  by  the  injection  of  cold  water  into 
the  cylinder  during  this  compression  stroke.  It  is 
interesting  at  least  to  compare  the  expression  for  the 
thermal  efficiency  of  this  cycle  with  those  already 
considered. 

The  method  of  obtaining  this  expression  is  of  the 
same  general  character  as  that  adopted  for  the  pre- 
vious cycles;  namely,  to  draw  the  cycle  in  the  tempera- 
ture entropy  plane  and  then  to  determine  the  heat 
quantities  as  represented  by  the  areas  under  the  proper 
lines.  The  only  new  feature  in  this  particular  formula 
is  the  heat  exhausted  during  the  isothermal  compres- 
sion. This  may  readily  be  obtained  by  multiplying 
the  temperature  of  exhaust  by  the  increase  in  entropy, 
which  may  be  determined  during  the  constant  volume 
generation  of  heat.  Going  through  the  steps  involved 
in  such  work  and  eliminating  the  temperatures  from 
the  formula,  as  in  the  preceding  cases,  we  arrive 
eventually  at  the  following  expression: 

Vol.  after  expansion  /    Q      \*-1 
10ge  Vol.  af tor  admissiorf      vol.      \ 


/Vol.  after  expansion^ fc  \  Vol.  af-  / 

\Vol.  after  admission/  V  tcr  ad./ 

This  expression  takes  a  general  form  similar  to  those 


204        THE  TEMPERATURE-ENTROPY  DIAGRAM  . 

of  the  preceding  cycles,  the  only  difference  being  that 
this  coefficient  is  smaller  than  any  of  the  preceding 
for  the  same  compression  and  tho  same  amount  of 
fuel.  We  further  see  that  the  denominator  increases 
more  rapidly  than  the  numerator  as  the  volume  after 
expansion  increases,  because  this  enters  the  denom- 
inator as  a  factor  proportional  to  its  k—l  powe'r,  while 
in  the  numerator  it  appears  simply  as  a  logarithm. 

An  instructive  tabulation  of  the  results  obtained 
from  these  different  cycles  may  be  made  by  writing 
them  in  the  general  form 

1  —  Eff.  =the  fractional  part  of  the  energy  wasted. 

ao- 

Diesel,     i-Jfe- 


/       Cl        X*-1 

otto,       i-^-ix(cl+RD  )    ; 


Cl.+P.D-ex 


fe-l) 

7C1.  +  P.D.1V 
\C1.+P.D.^/ 


Atkinson,    1  -BA  -k^  +  P.^ V     AcTTEDJ 

•ad 

Cl.+P.D.ex 


Max  1    F  0. 


CI.+P.D. 


GAS-ENGINE  CYCLES.  205 

If  the  Otto  cycle  be  taken  ac  the  standard  of  reference, 
the  waste  heat  of  the  other  cycles  may  be  expressed  in 
terms  of  the  waste  heat  of  the  Otto  by  means  of  the 
ratios 


Diesel  '~^~' 
Otto  i  ^.v_/. 
*l  ~CT~ 


Atkinson  Cl.  +P.D.ad 

Otto      =k~ 


-I 


/Cl.+P.D.ttA*       ' 
VCl.+PJD.J  ~ 


Maximum          -  Cl 


01  +P  D 
'  ' 


Otto 


ci.+p.D.exy-i_1  • 


CHAPTER  X. 

THE  GAS-ENGINE  INDICATOR  CARD. 

Physical  Conditions  of  the  Problem. — The  direct 
application  of  the  7^-analysis  to  an  actual  gas-engine, 
although  simple  in  theory,  is  difficult  in  practice.  To 
find  the  various  heat  losses  from  the  study  of  an  indi- 
cator card  necessitates  an  exact  knowledge  of  the  amount 
of  heat  available  per  revolution.  In  a  steam-engine 
this  may  be  determined  with  a  fair  degree  of  accuracy, 
provided  the  boiler  and  exhaust  pressures  remain  con- 
stant and  the  cut-off  varies  within  narrow  limits,  as 
then  each  pound  of  steam  contains  a  known  quantity 
of  heat  and  the  weight  required  to  develop  a  given 
power  is  measurable.  In  a  gas-engine  the  operation 
is  more  involved.  The  energy  is  not  drawn  from  a 
reservoir  but  is  generated  each  working  stroke  in  the 
cylinder.  Of  course  the  gas  can  be  metered  and  the 
average  consumption  found  per  working  stroke,  but 
how  is  the  corresponding  average  indicator  card  to  be 
determined?  It  is  easy  to  obtain  cards  from  the  same 
engine  showing  quick,  moderately  fast,  and  slow  burn- 
ing of  the  fuel.  (See  Figs.  81  and  82.)  How  are  the 

206 


THE  GAS-ENGINE  INDICATOR   CARD.  207 

corresponding  percentage  mixtures  and  the  absolute 
quantity  of  fuel  to  be  found  for  each  different  card? 
In  the  hit-or-miss  type  of  governing  the  "  miss  "  stroke, 
acfing  as  a  scavenger,  cools  the  cylinder  so  that  the 
following  charge  is  heated  less  during  admission,  is 
expanded  less,  and  hence  a  greater  weight  enters  the 
cylinder  than  would  directly  after  an  explosion.  The 
result  is  the  percentage  mixture  of  gas,  air,  and  burnt 
products  varies  throughout  the  explosions  for  a  com- 
plete firing  cycle.  Perhaps  the  best  that  can  be  done  is 
to  take  each  time  the  complete  set  of  cards  for  the 
firing  cycle  and  draw  an  equivalent  average  card.  If 
governing  is  effected  by  throttling  the  fuel  gas  there 
seems  to  be  no  method  of  determining  the  relative 
proportions  of  gas  and  air;  if  the  governor  controls 
either  by  throttling  or  by  cutting  off  the  admission  of  a 
mixture  of  known  composition,  the  relative  proportions 
between  it  and  the  burnt  products  will  still  be  un- 
known. Possibly  in  engines  of  the  Korting  type,  where 
the  gas  and  air  are  drawn  separately  into  cylinders  of 
known  capacity,  it  may  be  possible  to  attain  fairly 
accurate  knowledge  of  these  quantities  per  stroke. 

The  best  that  can  usually  be  done  is  to  keep  the 
load  as  constant  as  possible  and  use  average  cards  and 
average  fuel  consumption.  In  any  case  the  gas  and 
air  must  be  measured  independently  and  this  in  the 
case  of  a  large  gas-engine  will  require  a  large  gas-meter, 
or  some  form  of  displacement  tanks  for  measuring  the 


208       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

air.  If  nothing  else  is  at  hand,  an  anemometer  can  be 
used. 

Having  determined  the  average  indicator  card  and 
the  average  heat  applied  per  explosion,  the  next  diffi- 
culty is  the  determination  of  the  specific  heat  at  constant 

volume  and  the  ratio  —  .    The  difficulty  is  here  two- 

Cv 

fold:  first,  the  character  of  the  mixture  is  doubtful, 
due  to  the  varying  composition  of  the  burnt  gases  in 
the  clearance  space;  secondly,  our  knowledge  of  the 
variations  of  the  specific  heats  at  high  temperatures 
is  very  incomplete.  Thus  MM.  Mallard  and  Le  Chate- 
lier  found  by  experiment  the  values  : 

Carbonic  dioxide  ....   cv=  0.1477  +  0.000176* 

Water  vapor  ........      =0.3211  +  0.000219* 

Nitrogen  ...  .........      =  0.  170  +  0.0000872* 

Oxygen  ............      =0.1488  +  0.0000763* 

If  the  weight  of  each  component  present  in  the  charge 
is  known  the  value  of  cv  for  the  mixture  is  given  by 


Due  to  the  chemical  changes  during  combustion  the 
value  of  cv  differs  during  different  parts  of  the  cycle. 

The  value  of  k  is  likewise  a  variable.  Clerk  *  finds 
that  1.37  and  1.28  are  better  values  for  compression 

*  The  Engineer,  March  1  and  8,  1907,  pp.  205  and  236. 


THE  GAS-ENGINE   INDICATOR  CARD.  209 

and  expansion  respectively  than  the  customary  value 
1.4. 

The  specific  volume  of  the  charge  is  decreased  by 
combustion  due  to  the  rearrangement  of  the  molecules 
into  new  combinations.  Thus  for  some  samples  of 
Boston  illuminating-gas  the  ratio  of  final  to  initial 
specific  volumes  was  for  different  percentage  mixtures 
of  gas  and  air  as  follows: 

Gas  to  air 1:6          1:7          1:8          1:9 

Final  volume 


Initial  volume 


0.964      0.969      0.972      0.975 


Clerk*  states  that  1  volume  of  gas  and  5  of  air 
shrink  4  per  cent,  and  1  volume  of  gas  and  7  of  air 
shrink  3  per  cent,  and  1  volume  of  gas  and  10  of  air 
shrink  2.2  per  cent. 

To  transfer  the  gas-engine  indicator  card  to  the  T<j>- 
plane  requires  therefore 

(1)  Average  percentage  mixture  of  the  charge  and 
the  corresponding  average  card. 

(2)  The  laws  of  variation  of  cv  and  k. 

(3)  The  degree  of  combustion  so  that  the  shrinkage 
in  specific  volume  may  be  calculated. 

If  all  these  conditions  are  fulfilled  the  corresponding 
temperatures  and  entropies  can  be  calculated  and  the 
desired  plot  obtained. 

*  Min.  Proc.  Inst.  Civ.  Eng.,  vol.  CLXII,  p.  314. 


210       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

If  the  temperature  can  be  obtained  independently 
by  direct  measurement  for  any  point  of  the  cycle  it 
serves  as  a  check  upon  the  other  observations.  This 
would  require  some  thermo-electric  device  where  the 
circuit  is  closed  at  the  proper  moment  by  the  engine. 
The  difficulties  of  such  a  method  are  considerable,  and 
even  when  accurately  obtained  the  indicated  tempera- 
tures are  not  the  desired  average  temperature  but  the 
temperature  of  the  charge  at  the  point  at  which  the 
thermo-electric  couple  is  introduced.  Various  rough 
approximations  have  been  given  for  the  temperature 
of  the  charge  at  the  end  of  admission. 

Boulvin,  in  the  Entropy  Diagram,  assumes  it  to  be  a 
mean  between  that  of  the  surrounding  air  and  of  the 
jacket  water.  He  further  neglects  the  chemical  change 
due  to  combustion  and  assumes  that  the  values  of  cp, 
Cv,  and  k  of  the  charge  hold  sufficiently  well  for  the 
burnt  products,  and  further  assumes  that  cp  and  cv 
remain  constant  at  high  temperatures. 

Prof.  Burstall,  in  the  Second  Report  of  the  Gas-engine 
Research  Committee  (Proc.  Inst.  Mech.  Eng.,  1901, 
p.  1083),  takes  the  temperature  as  generally  not  differ- 
ing greatly  from  the  jacket  temperature.  He  finds  the 
computed  temperature,  however,  to  be  considerably 
higher  than  this  in  all  cases,  the  maximum  and  mini- 
mum differences  being  about  90°  and  32°  F.  He  also 
uses  the  variable  values  of  cv  given  on  page  175. 

Prof.  Reeve,  in  The  Thermodynamics  of  Heat-engines, 


THE  GAS-ENGINE  INDICATOR   CARD.  211 

states  that  he  usually  assumes  the  initial  temperature 
at  the  convenient  round  number  of  600°  absolute  Fah- 
renheit. 

Ideal  Indicator  Cards.— Having  settled  these  prelimi- 
naries, i.e.,  having  determined  the  average  card,  the 
average  heat  supplied  per  cycle,  the  values  of  cv  and  k 
for  the  mixture,  the  shrinkage,  and  the  initial  tempera- 
ture, the  next  step  is  to  draw  the  ideal  card. 
Thus  if  TQ = estimated  or  measured  temperature  at 

admission, 

po  =  initial  pressure  observed  from  card, 
^o  =  volume    of    clearance    plus    piston   dis- 
placement, 

1)I  =  volume  of  clearance, 
Qi  =  B.T.U.  received  per  cycle, 
Q2  =  B.T.U.  rejected  per  cycle, 

then,  referring  to  Fig.  73,  the  following  equations  give 
the  proper  values  of  the  coordinates  of  the  points  1, 
2,  and  3. 

A,  \  fc- 1 

Tl  =  TQ  ( — )      ,k  may  usually  be  assumed  as  1 .38 ; 


(See  p.  161.) 


212        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  ideal  cycle  will  thus  be  different  for  each  test 
as  in  each  case  the  gaseous  mixture  is  different,  so  that 
in  each  case  the  special  values  of  cv  and  k  must  be 
determined. 

This  is,  as  we  have  seen,  a  difficult  problem,  but, 
fortunately,  the  variations  in  cv  and  k  are  not  great 
between  different  tests  on  the  same  engine  and  between 
the  gaseous  mixtures  arising  from  different  fuels,  so 
that  these  variations  may  be  overlooked  without  intro- 
ducing too  great  an  error  in  the  results. 

Thus  Prof.  Lucke,  in  his  Gas-engine  Design,  establishes 
a  standard  reference  diagram  by  considering  the  work- 
ing fluid  to  be  air  filling  the  cylinder  at  the  end  of  the 
admission  stroke  under  14.7  Ibs.  pressure  and  at  32°  F. 
The  air  is  supposed  'to  pass  through  the  usual  Otto 
cycle,  during  ignition  receiving  at  constant  volume  all 
the  heat  generated  per  cycle  in  the  actual  engine. 
Such  a  device,  although  recommended  by  its  sim- 
plicity and  admirably  adapted  to  the  end  in  view,  viz., 
the  estabishment  of  diagram  factors  to  assist  in  the 
design  of  engines  of  any  required  power  and  working 
with  any  desired  fuel,  is  unsuitable  for  the  purposes  of 
heat  analysis. 

In  the  opinion  of  the  Committee  of  the  Institution  of 
Civil  Engineers  (see  Min.  Proc.,  1905,  pp.  324  and  326), 
"it  would  introduce  some  uncertainty  and  difficulty, 
without  adequate  compensating  advantage,  to  make 
the  standard  engine-cycle  depend  on  a  knowledge  of 


THE  GAS-ENGINE  INDICATOR  CARD  213 

the  physical  constants  of  the  exact  mixture  used.  The 
discussion  of  the  constant  for  various  mixtures  of  gases 
already  given  shows  that,  apart  from  the  unknown 
change  at  high  temperatures,  these  constants  do  not 
differ  by  more  than  2  per  cent,  to  5  per  cent,  from  those 
of  air  in  such  mixtures  as  are  used  in  gas-engines.  The 
advantages  of  simplicity  and  definiteness  in  the  standard 
are  so  great  that  the  Committee  recommend  that  the 
standard  engine  should  be  taken  to  work  with  a  perfect 
gas  of  the  same  density  as  air.  This  in  no  way  prevents 
any  one  from  discussing  the  distribution  of  heat  losses 
in  any  particular  trial  with  constants  adjusted  -to  any 
particular  mixture  of  gases.  But  it  does  render  more 
definite  the  statement  of  the  relative  efficiency  (cylinder 
efficiency),  without,  in  the  opinion  of  the  Committee, 
introducing  any  error  of  practical  importance. 

"  The  standard  engine  of  comparison  is  therefore  a 
perfect  air-gas  engine  operated  between  the  same 
maximum  and  minimum  volumes  as  the  actual  engine, 
receiving  the  same  total  amount  of  heat  per  cycle,  but 
without  jacket  or  radiation  loss,  and  starting  from  one 
atmosphere  and  the  selected  initial  temperature  of 
139°  F.  Its  efficiency  is  the  same  as  that  of  a  Carnot 
engine  working  through  the  temperature  range  T\  —  TQ 
or  T2  —  T3  (Fig.  80).  The  efficiency  is  given  by  a  very 
simple  expression,  depending  only  on  the  dimensions 
of  the  cylinder  and  independent  of  the  heat  supply  or 
the  maximum  temperature.  If  the  heat  supply  is 


214       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


increased  by  using  a  richer  charge,  the  work  done  is 
increased,  but  not  the  efficiency.  The  pressure-volume 
or  temperature-entropy  diagram  of  the  standard  cycle 
can  be  drawn  on  the  assumptions  just  stated,  which,  in 
the  opinion  of  the  Committee,  do  not  essentially  differ 
from  those  in  actual  engines." 

The  only  effect  of  using  the  "  standard  air  cycle  " 
instead  of  the  actual  ideal  cycle  will  be  in  each  case  to 
change  the  values  of  the  ideal  and  cylinder  efficiencies 
a  few  per  cent.  Thus  the  ideal  efficiency  varies  as 
shown  in  the  following  table: 


(Clearance  +  piston 
displacement) 

l     /                      clearance                     \k-l 

^Clearance  +  piston  displacement  / 

fc=1.40 

k=l.37 

2 

0.246 

0.225 

3 

0.36 

0.332 

4 

0.43 

0.399 

5 

0.47 

0.45 

7 

0.55 

0.51 

10 

0.61 

0.57 

20 

0.70 

0.67 

100 

0.85 

0.82 

In  ordinary  commercial  gas-engines  of  moderate 
size,  the  cylinder  efficiencies,  referred  to  the  air-engine 
standard,  vary  from  0.5  to  0.6.  A  comparision  of  the 
efficiencies  of  actual  engines  and  the  air-engine  standard 
is  given  in  Mr.  Dugald  Clerk's  paper,  "Recent  Develop- 
ments in  Gas-engines,"  in  Min.  Proc.  Inst.  C.  E.,  Vol. 
cxxiv,  p.  96. 


THE  GAS-ENGINE  INDICATOR  CARD. 


215 


In  discussing  the  effect  of  size,  speed,  clearance,  time 
of  firing,  etc.,  upon  engines  using  the  same  fuel,  the 
air-standard  cycle  can  be  used  to  good  advantage, 
but  in  studying  the  heat  losses  due  to  different  fuels, 
etc.,  an  error  is  liable  to  be  introduced  unless  the  proper 
value  of  k  is  used  for  each  mixture. 


ir  o 


FIG.  80. 

Cylinder  Efficiency.— The  discrepancy  between  the 
ideal  and  the  actual  cards  is  due  to  three  influences : 

First,  the  shrinkage  due  to  combustion  would  cause 
the  expansion  line  of  the  ideal  card  to  occur  at  pressures 
reduced  proportionally  to  its  magnitude.  Less  work 
is.  thus  done  during  the  expansive  stroke  while  the 
work  of  compression  is  undiminished,  thus  reducing 
the  net  work  obtainable  from  the  cycle.  This  new 


216       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

expansion  line  would  occupy  some  such  position  as  xx 
That  portion  of  the  ideal  card  beyond  xx  is,  therefore 
from  the  nature  of  the  substance  unattainable.  Fig.  80. 

Next,  there  is  the  loss  due  to  incomplete  combustion. 
This  may  be  caused  by  a  mixture  of  such  proportions 
that  the  fuel  cannot  all  burn,  or  by  one  in  which  the 
combustion  is  retarded  so  much  as  to  be  unfinished  at 
exhaust;  in  either  case  the  full  heat  value  of  the  charge 
is  not  developed  in  the  cylinder.  Although  such  a 
charge  does  not  burn  at  constant  volume,  but  is  burning 
throughout  part  or  all  of  the  expansive  stroke,  and 
cannot  thus  be  represented  on  the  ideal  cycle,  yet  the 
magnitude  of  the  lost  heat  may  be  represented  by 
drawing  yy  so  that  the  area  between  xx  and  yy  equals 
this  loss.  The  area  under  ly  is  thus  equal  to  the  heat 
actually  generated  in  the  cycle. 

Finally,  there  is  the  loss  due  to  heat  interchange 
between  the  gas  and  the  metal  which  produces  the 
discrepancy  between  that  portion  of  the  ideal  card  to 
the  left  of  yy  and  the  actual  card. 

It  is  almost  never  possible  to  locate  xx  and  yy,  as  the 
amount  of  shrinkage  in  the  cylinder  has  to  be  estimated 
from  the  analysis  of  the  burnt  gases  in  the  exhaust- 
pipe.  But  with  delayed  combustion  the  gas  may  not 
be  the  same  in  the  two  places.  The  best  that  can  be 
done  is  to  draw  xx  to 'correspond  to  the  shrinkage  for 
complete  combustion. 

The  losses  due  to  incomplete  combustion  and  to  heat 


THE  GAS-ENGINE   INDICATOR  CARD.  217 

interchanges  with  the  cylinder  walls  are  thus  com- 
bined, due  to  our  inability  to  locate  yy.  But  by  suitably 
regulating  the  percentage  mixture  it  is  possible  nearly 
to  eliminate  the  loss  due  to  incomplete  combustion, 
although  •  as  the  governor  controls  the  percentage 
mixture  either  of  air  and  gas,  or  of  mixture  and  burnt 
products,  slow-burning  charges  sometimes  result. 

The  cylinder  efficiency,  i.e.,  the  ratio  of  the  actual 
to  the  ideal  card,  is  thus  limited  by  two  different  factors, 
the  physical  properties  of  the  fuel — shrinkage  and  rate 
of  flame  propagation — and  the  influence  of  the  cylinder 
walls.  It  is  in  tracing  out  the  heat  interchanges  be- 
tween the  charge  and  the  cylinder  waUs  that  the  T(j>- 
analysis  is  of  greatest  value. 

Actual  Indicator  Cards. — To  plot  the  ^-projection 
of  a  gas-engine  indicator  card  the  temperature  and 
entropy  for  a  sufficient  number  of  points  can  be  calcu- 
lated from  the  equations 


10g«? 


=  cv  loge  PX+CP  loge  vx— constant, 

where  Tx  and  $x  of  the  desired  point  are  expressed 
in  terms  of  the  observed  pressures  and  volumes  as 
measured  on  the  indicator  card. 


218       THE  TEMPERATURE -ENTROPY  DIAGRAM. 


This  computation  presupposes  a  knowledge  of   T0, 
but  if  this  is  unknown  it  can  be  assumed,  and  then  the 


the  7^-plot  will  give  relative  values  of  T  and  <£  but 
not  absolute  values.  As  this  operation  must  be  re- 
peated for  a  number  of  points  large  enough  to  deter- 


THE  GAS-ENGINE  INDICATOR  CARD. 


219 


mine  the  curves  with  sufficient  accuracy,  it  requires 
much  time  and  patience. 


The  method  of  graphical  projection  from  the  pv- 
into  the  !T<£-plane,  as  explained  in  Chapter  II,  was 
used  in  drawing  Figs.  81,  82,  and  83.  The  indicator 


220       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


cards  were  taken  from  the  36  horse-power  gas-engine 
at  the  Institute:  Boston  illuminating  gas  being  used 
as  fuel.  Fig.  81  represents  an  average  card  with  a  good 


Atmospheric 


FIG.   83. — Card   taken  with    weak    spring,   showing     compression 
line,  and  negative  work  during  admission  and  exhaust. 

mixture  and  well  timed  ignition,  while  Fig.  82  shows 
the  result  of  using  a  weak  mixture  and  late  ignition. 
Fig.  83  was  taken  with  a  weak  spring  to  show  more 
clearly  the  character  of  the  exhaust,  admission,  and 
compression  strokes.  The  small  oscillations  are  chiefly 
due  to  the  weak  spring. 


THE  GAS-ENGINE   INDICATOR   CARD.  221 

Heat    Interchanges    between    Gas    and    Cylinder. — 

There  is  a  continuous  transmission  of  heat  through  the 
cylinder  walls  to  the  jacket  water  or  to  the  air-cooling 
surface  which  may  amount  to  from  25  per  cent,  to 
40  per  cent,  of  the  heat  of  combustion.  This  prevents 
the  temperature  and  pressure  in  the  cylinder  from  ever 
attaining  their  theoretical,  values. 

Besides  this  heat  which  is  thus  lost  there  is  a  second 
quantity  which  passes  first  to  the  walls  and  then  back 
to  the  gas.  We  might  perhaps  expect  to  find  this 
small  compared  with  the  condensation  and  re-evap- 
oration in  a  steam-cylinder  due  to  the  less  rapid  inter- 
change of  heat  between  gas  and  metal  than  between 
wet  vapor  and  metal,  but  in  the  gas-engine  we  must 
remember  that  we  are  dealing  with  much  greater  tem- 
perature differences. 

The  compression  line  in  the  T^-plane  slants  usually 
first  to  the  right,  showing  the  transference  of  heat 
from  the  hot  metal  to  the  cold  charge.  Continued 
compression,  however,  raises  the  temperature  of  the 
charge  until  it  becomes  equal  to  the  mean  tempera- 
ture of  the  cylinder  walls.  Beyond  this  point  the  com- 
pression curve  slants  to  the  left,  as  the  flow  of  heat  is 
now  from  the  gas  to  the  metal.  As  the  piston  approaches 
the  end  of  the  compressive  stroke  its  speed  is  slow, 
so  that  the  rate  of  increase  of  tempeature  due  to  com- 
pression is  slow,  but  the  temperature  difference  between 
gas  and  metal  being  large  the  loss  of  heat,  arid  hence 


222       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

rate  of  temperature  decrease,  due  to  heat  transference, 
is  large.  Thus  the  compression  curve  approaches  more 
and  more  closely  to  the  isothermal  as  the  conduction 
loss  approaches  the  compression  gain.  It  may  some- 
times happen  that  the  loss  exceeds  the  gain  and  then 
the  temperature  actually  decreases  towards  the  end  of 
compression.  The  magnitudes  of  the  heat  transfer- 
ence for  any  case  may  be  determined  by  planimetering 
the  areas  under  the  respective  curves  and  expressing 
in  heat  units. 

If  the  ignition  occurs  before  the  end  of  the  com' 
pression  stroke,  and  if  the  combustion  is  not  finished 
until  after  the  piston  has  passed  the  dead-center,  the 
combustion  line  at  first  approaches  the  constant-volume 
curve  of  the  ideal  cycle,  becomes  tangent  to  it  at  the 
moment  dead-center  is  passed,  and  then  proceeds  to 
fall  off  from  it  until  combustion  is  finished.  Figs.  81 
and  82  show  the  character  of  the  combustion  curve 
for  different  times  of  firing. 

The  expansion  line  varies  widely  in  character,  accord- 
ing to  the  percentage  mixture  of  the  charge,  the  time 
of  firing,  and  the  speed  of  the  engine.  For  slow- 
burning  mixtures  (slow  burning  with  reference  to  the 
time  duration  of  a  revolution)  the  line  may  slope  con- 
tinuously to  the  right,  showing  the  constant  addition 
of  heat  up  to  the  moment  of  release  (Fig.  82).  Mix- 
tures in  which  the  combustion  is  finished  shortly  after 
the  beginning  of  expansion  give  cards  of  approximately 


THE  GAB-ENGINE  INDICATOR  CARD.  223 

the  character  shown  in  Fig.  81.  At  first  the  expansion 
line  frequently  shows  decreasing  entropy  due  to  the 
rapid  interchange  of  heat  between  the  incandescent 
gas  and  the  relatively  cold  metal.  The  inner  layer  of 
metal  soon  reaches  the  temperature  of  the  gas  and  then 
as  the  temperature  of  the  gas  is  decreased  by  further 
*  expansion  the  inner  surface  of  the  cylinder  possesses  the 
highest  temperature  and  heat  flows  from  it  in  two 
directions,  outwards  towards  the  cold  jacket  and  in- 
wards to  the  cold  gas.  This  last  change  manifests 
itself  by  increasing  entropy  during  the  latter  part  of 
the  expansion.  Sometimes  this  second  part  of  the 
expansion  line  becomes  practically  isothermal  and  thus 
lends  credence  to  the  theory  of  "  after  burning  " — this 
isothermal  process  being  apparently  at  the  highest 
temperature  at  which  recombination  of  the  dissociated 
elements  may  occur. 

The  character  of  the  exhaust  line  is  of  no  significance, 
as  it  does  not  represent  the  history  of  a  fixed  quantity 
of  substance.  Its  sole  importance  is  to  close  the  dia- 
gram and  thus  to  make  the  area  of  the  T^-diagram  the 
heat  equivalent  of  the  work  recorded  by  the  indicator 
card. 


CHAPTER  XI. 


THE    TEMPERATURE-ENTROPY    DIAGRAM    APPLIED 
TO  THE  NON-CONDUCTING  STEAM-ENGINE. 

THE  Carnot  cycle  for  steam  gives  a  very  good  pv-d\a- 
gram,  and  hence  there  are  not  the  same  mechanical 
objections  to  its  adoption  as  in  hot-air  engines.  But, 
due  to  the  physical  change  in  the  working  fluid,  a  differ- 


FIG.  S4. 


ent  cycle  has  proved  to  be  more  feasible.  In  the 
Carnot  engine  the  steam  at  condition  d,  Fig.  84,  would 
be  compressed  adiabatically  to  a  with  the  change  in 


224 


NON-CONDUCTING  STEAM-ENGINE. 


225 


condition  from  xd  to  xa.  The  isothermal  expansion 
ab  occurring  by  the  application  of  heat  to  the  cylinder 
produces  the  further  change  in  condition  to  x&.  The 
cycle  is  finished  by  the  adiabatic  expansion  be  and 
the  isothermal  compression  cd  with  the  cylinder  in  con- 
tact with  the  refrigerator. 

The  card  of  the  ideal  engine  differs  materially  from 
this.     (1)  The  line  ab,  Fig.  85,  represents  the  admission 


a         1,1 


FIG.  So. 


of  steam  of  condition  xi  into  the  cylinder  up  to  the 
point  of  cut-off.  This  is  forced  in  by  the  vaporiza- 
tion of  an  equivalent  amount  in  the  boiler,  so  that  the 
T<£-curve  is  the  same  as  in  the  Carnot  engine.  (2)  The 
adiabatic  line  be  represents  the  expansion  of  the 
steam  admitted  along  ab  plus  the  amount  already 
in  the  clearance  space  at  the  moment  of  admission  a. 
(3)  During  exhaust  the  piston  simply  forces  out  into  the 


226       THE   TEMPERATURE-ENTROPY  DIAGRAM. 

condenser  all  of  the  steam  taken  during  admission,  but 
the  quality  of  the  remaining  portion  is  the  same  at 
compression  as  at  the  beginning  of  release.  (4)  The 
part  confined  in  the  clearance  space  is  then  compressed 
along  da  in  the  pr-plane  up  to  the  initial  pressure, 
i.e.,  back  along  the  curve  cb  in  the  7^-plane.  Hence, 
in  a  non-conducting  engine,  the  amount  of  steam  con- 
fined in  the  clearance  space  is  immaterial,  as  its  expan- 
sion and  compression  occur  along  the  same  adiabatic 
and  do  not  affect  the  heat  consumption. 

That  part  of  the  steam  exhausted  during  release, 
however,  passes  into  the  condenser  and  there  con- 
denses and  gives  up  its  heat  to  the  cooling  water. 
This  is  represented  by  cd. 

From  the  condenser  the  water  is  forced  into  the 
boiler  by  means  of  a  feed-pump,  and  is  there  warmed 
from  d  to  a  and  vaporizes  from  a  to  6.  The  pv-di&- 
gram  gives  a  history  of  the  work  performed  per  stroke 
and  is  confined  entirely  to  the  events  in  the  cylinder. 
The  !T</>-diagram,  however,  represents  the  heat  cycle, 
and  consists  of  events  occurring  ia  three  different 
places,  da  and  ab  represent  the  heating  of  the  feed- 
water  and  its  evaporation  at  working  pressure  in  the 
boiler,  be  represents  the  adiabatic  expansion  in  the 
cylinder  of  the  engine,  and  cd  the  discharge  of  heat 
to  the  condenser. 

If  it  were  desired  to  make  this  cycle  into  a  Carnot> 
the  condensation  would  have  to  stop  at  df  and  the 


NON-CONDUCTING  STEAM-ENGINE.  227 

feed-pump   arranged   to   compress   the   mixture   adia- 
batically  to  a. 

Suppose  each  engine  to  use  one  pound  of  steam  of  con- 
dition x^  Fig.  85,  per  stroke,  then  the  efficiency  will  be 

0)    Carnot:      ^"^I^'^l 

area  abnm  —  area  d'cnm     Tl—T2  t 
area  abnm  Tl 

6)     Non-conducting  or  Rankine  engine: 


ql+xbrl-qz- 


An  inspection  of  the  diagram  shows  at  once  that 

^  Carnot    >    ^  Rankine' 

It  is  evident  also  that  for  any  given  boiler  pressure, 
the  less  the  amount  of  moisture  in  the  steam  the  smaller 
the  difference  between  the  Carnot  and  the  Rankine 
cycles. 

Increased  Efficiency  by  Use  of  High-pressure  Steam.  — 
If  the  same  quantity  of  heat  be  supplied  per  pound 
of  steam  under  constantly  increasing  pressure  the 
state  point,  6,  Fig.  SO,  will  assume  the  successive  posi- 


228       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


tions  bf,  b",  etc.,  along  the  constant  heat  curve  W, 
and  at  the  same  time  the  state  point  c,  representing 
the  condition  at  the  end  of  the  adiabatic  expansion  to 
the  back  pressure,  moves  towards  the  left  into  the 
successive  positions  c',  c",  etc.  Now  the  areas  under 
cd,  c'd,  c"d,  etc.,  represent  the  quantity  of  heat  dis- 


FIG.  83. 

charged  to  the  condenser  under  the  different  condi- 
tions. Therefore  the  greater  the  pressure  or  the  higher 
the  temperature  at  which  a  given  quantity  of  heat  is 
supplied  to  the  engine,  the  smaller  the  fractional  part 
rejected  to  the  condenser,  that  is,  the  larger  the  por- 
tien  turned  into  work  and  the  greater  the  efficiency. 
The  Tp-curvc  shows  that  at  high  pressures  the 
pressure  increases  much  more  rapidly  than  the  tem- 
perature, and  hence  the  necessary  strength,  weight, 
and  cost  of  the  engine  will  increase  more  rapidly  than 
the  gain  in  efficiency. 


THE  NON-CONDUCTING  STEAM-ENGINE.         229 

It  should  be  noted,  however,  that  there  is  also  an 
upper  limit  to  the  theoretical  gain  in  efficiency  from 
increasing  the  initial  pressure.  Although  our  knowledge 
of  the  properties  of  saturated  steam  above  310  pounds 
pressure  is  inexact  and  largely  obtained  by  extra- 
polation, nevertheless  it  suffices  to  define  approxi- 
mately the  contour  of  the  water  and  steam  lines  up  to 
their  junction  at  the  critical  temperature  (see  table  on 
p.  388).  The  complete  diagram  for  saturated  steam  is 
approximately  as  shown  in  Fig.  87.  There  is  no  doubt 
that  for  a  while  increasing  the  pressure  increases  the 
efficiency,  but  at  the  same  time  the  heat  of  vaporiza- 
tion is  diminishing  with  increasing  rapidity  (approach- 
ing zero  as  a  limit  at  the  critical  temperature),  so  that, 
as  the  water  and  steam  lines  converge,  the  discrepancy 
between  the  Rankine  and  the  Carnot  cycles  grows  more 
and  more  marked.  From  being  nearly  identical  at 
lower  pressures  the  Rankine  efficiency  finally  attains 
to  only  about  one  half  the  Carnot  efficiency.  For  dry 
steam  the  point  at  which  the  Rankine  efficiency  reaches 
its  maximum  value  appears  to  be  at  about  a  pressure 
of  2000  pounds,  the  exact  point  varying  somewhat 
with  the  back  pressure. 

Table  of  Rankine  Efficiencies. — The  table  of  efficiencies 
on  page  231  shows  very  clearly  the  relative  gains 
to  be  expected  from  increase  of  boiler  pressure  or 
decrease  of  back  pressure.  Thus  starting,  for  example, 
with  an  absolute  boiler  pressure  of  70  pounds  and  run- 


230        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


-|  500 

I450 
ft  428 
2  400 


•HSOO 
2 
5  250 

8.200 

H  150 

100 

50 
33 
0 


CriticaHemperatnre^ 


Range  of  Peabody's  Steam  Tables 


0.6       0.8        1.0       1.2       1.4 
Units  of  Entropy. 

FlO.  87. 


1.6       1.8       2.0       2.2 


THE  NON-CONDUCTING  STEAM-ENGINE.        231 


RANKINE'S  EFFICIENCY  WITH  DRY  SATURATED  STEAM 
FOR  DIFFERENT  INITIAL  AND  FINAL  PRESSURES. 


Initial  Conditions. 

Back  Pressures,  Pounds  Absolute. 

pi  • 

h 

20.0 

14.7 

5 

4 

3 

2 

l 

5.99 

170 

0.3 

1.7 

3.4 

5.9 

9.7 

7.51 

180 

2.7 

4.1 

5  9 

8  2 

11  8 

9^34 

190 

4.2 

5.6 

7.3 

9.6 

13.1 

11.53 

200 

5.6 

6.9 

8.6 

10.8 

14.3 

14.13 

210 

.... 

6.9 

8.2 

9.9 

12.1 

15.5 

14.70 

212 

7.2 

8.5 

10.1 

12.4 

15.6 

17.19 

220 

i.2 

8.3 

9.5 

11.1 

12.7 

16.6 

20.78 

230 

'6!3 

2.6 

9.5 

10.8 

12.3 

14.4 

17.6 

25.0 

240 

1.7 

3.9 

10.7 

11.9 

13.4 

15.4 

18.6 

29.8 

250 

3.1 

5.3 

11.8 

13.0 

14.4 

16.5 

19.6 

35.4 

260 

4.4 

6.5 

12.9 

14.1 

15.6 

17.5 

20.6 

41.8 

270 

5.6 

7.7 

14.1 

15.2 

16.6 

18.5 

21.5 

49.2 

280 

6.9 

8.9 

15.1 

16.2 

17.6 

19.4 

22.4 

57.5 

290 

8.2 

10.0 

16.0 

17.1 

18.5 

20.4 

23.2 

67.0 

300 

9.1 

11.1 

17.0 

18.1 

19.5 

21.2 

24.0 

77.6 

310 

10.2 

12.1 

17.9 

19.0 

20.3 

22.1 

24.8 

89.6 

320 

11.3 

13.2 

18.8 

19.9 

21.1 

22.8 

25.7 

103.0 

330 

12.3 

14.1 

19.7 

20.7 

21.9 

23.7 

26.3 

117.9 

340 

13.2 

15.1 

20.5 

21.5 

22.7 

24.4 

27.0 

134.5 

350 

14.2 

15.9 

21.3 

22.3 

23.5 

25.1 

27.7 

152.9 

360 

15.1 

16.9 

22.0 

23.0 

24.2 

25.8 

28.4 

173.2 

370 

16.0 

17.7 

22.8 

23.7 

25.0 

26.5 

29.0 

195.5 

380 

16.8 

18.5 

23.5 

24.4 

25.5 

27.2 

29.6 

220.0 

390 

17.6 

19.2 

24.2 

25.1 

26.2 

27.8 

30.2 

246.9 

400 

18.4 

20.0 

24.8 

25.8 

26.9 

28.4 

30.7 

276.3 

410 

19.1 

20.8 

25.5 

26.4 

27.5 

29.0 

31.3 

308  .  5 

420 

19.8 

21.4 

26.1 

27  0 

28.1 

29.6 

31.8 

336.2 

428 

20.4 

22.0 

26.6 

27.5 

28.6 

30.0 

32.2 

422.4 

450 

21.9 

23.3 

27.9 

28.7 

29.7 

31.1 

33.3 

678.5 

500 

24.8 

26.2 

30.6 

31.1 

31.2 

33.7 

35.4 

9.-)()  .  1 

540 

27.3 

28.0 

31.9 

32.7 

33.6 

34.8 

36.7 

1212 

570 

28.0 

29.2 

32.9 

33.6 

34.5 

35.7 

37  .  5 

1516 

600 

28.9 

30.0 

33.6 

34.3 

35.2 

36.3 

38.1 

1867 

630 

29.5 

30.6 

34.1 

34.7 

35.5 

36.6 

38.4 

2137 

650 

29.6 

30.7 

34.1 

34.7 

35.5 

36.6 

38.3 

2431 

670 

29.3 

30.3 

33.6 

34.3 

35.0 

36.0 

37.7 

2748 

690 

21.4 

22.7 

26.8 

27.4 

28.4 

29.6 

31.6 

2882 

698 

20.8 

22.0 

25.6 

26.3 

27.1 

28.2 

30.1 

232        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

ning  non-condensing  the  Rankine  efficiency  is  found 
to  be  11.4  per  cent.  If  the  back  pressure  is  lowered 
to  1  pound  absolute  the  efficiency  becomes  24.4  per 
cent  or  is  more  than  doubled,  while  if  the  initial  pressure 
is  raised  to  330.2  pounds  absolute  the  efficiency  becomes 
only  22.0  per  cent.  The  combination  of  both  changes 
results  in  an  efficiency  of  32.2  per  cent.  Thus  in  the 
case  cited  decreasing  the  back  pressure  13.7  pounds 
produces  a  greater  gain  than  that  obtained  by  increas- 
ing the  initial  pressure  by  266  pounds.  This  illustrates 
the  great  gain  in  efficiency  to  be  obtained  by  running 
an  engine  condensing. 

The  Low-pressure  Turbine.  —  Steam  exhausted  at 
atmospheric  pressure  is  still  capable  of  developing 
15.6  per  cent  of  its  total  heat  into  work  if  used  in  a 
turbine  exhausting  at  1  pound  absolute,  which  is  equal 
to  the  work  to  be  obtained  from  a  non-condensing  engine 
taking  steam  at  128.3  pounds  absolute. 

To  utiMze  to  full  advantage  the  heat  in  low  pressure 
steam,  a  reciprocating  engine  would  need  to  be  exceed- 
ingly large  and  thus  possess  large  friction  losses,  where- 
as a  turbine  is  especially  adapted  for  low  pressures  as 
a  large  part  of  its  friction  is  due  to  the  windage  of  the 
rotor  and  this  diminishes  as  the  density  of  the  working 
fluid  diminishes.  The  ideal  unit  for  economical  power  pro- 
duction would  therefore  seem  to  be  a  reciprocating  engine 
for  the  high  pressures  combined  with  a  turbine  for  the  low 
pressures,  thus  combining  the  best  conditions  for  both. 


NON-COXDUCTIXG  STEAM-ENGINE.  233 

That  is,  since  the  clearance  between  the  blades  and 
guides  of  a  turbine  must  possess  the  same  absolute 
value  in  all  stages,  the  clearance  area  relatively  to  the 
blade  area  must  be  larger  in  the  high  pressure  stages 
than  in  the  low,  so  that  from  this  view  point  alone  the 
efficiency  of  the  high  pressure  stages  might  be  expected 
to  be  less  than  that  of  the  low  pressure  stages.  But 
this  is  just  the  condition  under  which  steam  can  be 
most  advantageously  used  in  a  reciprocating  engine 
as  its  small  volume  permits  of  small  cylinders  and  high 
mean  effective  pressures. 

Thus  a  combined  unit  using  a  reciprocating  engine 
where  the  turbine  is  least  efficient  and  using  a  turbine 
where  the  reciprocating  engine  is  least  efficient,  may  be 
expected  to  give  a  higher  total  thermal  efficiency  than 
either  type  of  motor  used  alone  throughout  the  entire 
range  of  pressure. 

Effect  of  Different  Substances  upon  the  Rankine 
Efficiency. — The  discrepancy  between  the  Rankine  and 
Carnot  cycles  is  caused  by  the  slope  of  the  liquid  line. 
The  larger  the  specific  heat  of  the  liquid  the  greater 
the  growth  of  entropy  during  rise  of  temperature  and 
the  more  the  liquid  line  inclines  from  the  isentropic. 
That  is,  other  things  being  equal,  the  substance  possess- 
ing the  smallest  specific  heat  would  give  the  maximum 
value  of  the  Rankine  efficiency.  (See  LI  and  Z/2, 
Fig.  88.) 

The  effect  of  the  sloping  liquid  line  is  further  enhanced 


234        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


or  diminished  by  the  relative  positions  of  the  liquid  and 
dry  vapor  lines,  V\  and  V2,  Fig.  83.  Thus  if  the  latent 
heat  of  vaporization  were  doubled  the  discrepancy 
between  the  Carnot  and  Rankine  cycles  would  for  the 
same  specific  heat  of  the  liquids  be  about  halved,  etc. 

That  substance  possessing  the  smallest  specific  heat 
and  the  largest  latent  heat  of  vaporization  will  give  the 
maximum  value  of  the  Rankine  efficiency  between  any 
two  temperatures.  The  vapor  tables  show  that  these 
properties  are  varying  simultaneously,  so  that  it  becomes 


\       \ 


FIG.  88. 

a  special  problem  for  any  given  temperature  range  to 
determine  which  of  the  available  fluids  would  give  the 
maximum  Rankine  efficiency. 

For  the  engineer,  however,  the  thermal  efficiency  is 
but  one  of  many  factors.  Thus  if  the  same  power  is  to 
be  developed  with  the  different  fluids  a  different  weight 
of  each  substance  must  be  used,  so  that  w(Hi  —  H2}  may 
be  the  same.  This  means  a  different  pressure  and 
volume  range,  and  thus  the  size  and  strength  of  the 


NON-CONDUCTING  STEAM-ENGINE. 


235 


engine  to  be  used  with  each  fluid  would  need  to  be 
computed.  Finally  the  cheapness  and  availability  of 
each  must  be  considered. 

Gain  in  Efficiency  from  Decreasing  the  Back  Pressure. 
— If  the  initial  pressure  be  kept  constant  (Fig.  89)  and 
the  back  pressure  be  diminished  by  increasing  the 
vacuum,  the  heat  taken  up  in  the  boibr  by  jach  pound 


m,   ,m 


FIG.  89. 

of  steam  will  be  increased  from  dabnm  to,  say,  d'dbnm', 
and  the  heat  discharged  to  the  condenser  will  diminish 
from  dcnmto  d'c'nm';  that  is,  the  efficiency  increases. 

Again  referring  to  the  pT-curve,  it  is  clear  that  at  low 
pressure  the  temperature  decreases  much  more  rapidly 
than  the  pressure,  so  that  a  small  decrease  in  pressure 
means  a  considerable  increase  in  efficiency.  This  is  at 
once  evident  from  an  inspection  of  the  efficiency  for  a 


Carnot  cycle,  y  =  — 
• 

the  Rankine  cycle, 


''•  =l-^r,  but  the  expression  for 


236       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

does  not  show  the  influence  of  .the  upper  or  lower 
temperature  very  clearly.  The  efficiency  may  easily 
be  expressed  as  a  function  of  the  upper  and  lower 
temperatures  by  assuming  the  value  of  the  specific 
heat  of  water  to  be  constant  and  equal  to  unity.  Thus 


and 


The  value  of  the  last  term  decreases,  hence  the 
efficiency  increases, 

(1)  as  xb  approaches  unity, 

(2)  as  7\  increases,  and 

(3)  as  T2  decreases. 

Gain  in  Efficiency  from  Using  Superheated  Steam. — 
To  avoid  the  introduction  of  excessively  high  pressures 
superheated  steam  is  being  used  more  and  more.  Accord- 
ing to  the  Carnot  cycle  the  gain  in  efficiency  is  equally 
great  whether  superheated  or  saturated  steam  of  the 


NON-CONDUCTING  STEAM-ENGINE. 


237 


same  temperature  is  used,  but  the  Rankine  cycle  shows 
that  the  theoretical  gain  to  be  expected  from  super- 
heated steam  is  but  slight. 

The  portion  be  of  the  Rankine  cycle,  Fig.  90.  repre- 
sents the  addition  of  heat  in  the  superheater,  and  ec 
the  expansion  from  superheated  to  saturated  steam 


— 


FIG.  90. 

in  the  cylinder;  the  rest  of  the  cycle  is  as  previously 
described. 

The  heat  gx  —  52+ri  +CP(^~^I)  is  received  along  the 
line  of  varying  temperatures  dabe,  while  in  the  Carnot 
cycle  an  equal  quantity  of  heat  (area  e//Zn  =  area  dabenrn) 
is  all  received  at  the  upper  temperature  t8.  Hence  the 
efficiency  of  the  Rankine  is  now  much  less  than  that 
of  the  Carnot  cycle  working  between  the  same  tem- 
perature limits  and  the  discrepancy  increases  as  the 
degree  of  superheating  increases. 


23S       THE  TEMPERATURE-ENTROPY  DIAGRAM. 
The  analytical  formulae  for  this  case  are: 

cp(tg-t1]  -xcr2 xcr2 


7\     r,  TV 

=  1  —  -^Tf, — m       l TTf, — 7fr\~  (approximately). 

This  shows  an  increase  in  efficiency  "with  increasing 
Ta,  but  only  of  small  amount. 

It  is  evident,  then,  that  the  great  gain  obtained  by 
using  superheated  steam  must  be  looked  for  in  the 
overcoming  of  certain  defects  inherent  in  an  actual 
engine.  The  use  of  steam  expansively  entails  a  cool- 
ing of  the  working  fluid  and  hence  of  the  cylinder  walls 
containing  it.  This  effect  is  increased  by  release 
occurring  before  the  expansion  has  reached  the  back 
pressure,  and  is  only  partially  counteracted  in  part 
of  the  cylinder  walls  by  the  heating  effect  produced 
during  compression.  Thus  the  entering  steam  under- 
goes partial  condensation  before  the  cylinder  walls 
have  been  brought  up  to  its  temperature;  that  is, 
each  pound  of  steam,  instead  of  occupying  the  volume 
which  it  had  in  the  steam-pipe,  now  occupies  a  reduced 
volume  proportional  to  the  condensation.  And  hence, 
instead  of  obtaining  the  total  area  abed,  Fig.  91,  only 
the  fractional  part  akld  can  be  utilized.  Thus  the 
area  kbcl  has  been  subtracted  from  the  numerator  of 
the  expression  for  efficiency.  This  condensation  may 


NON-COND  VOTING  STEA M-ENGINE. 


239 


range  as  high  as  from  20  per  cent,  to  50  per  cent,  of 
the  total  steam. 

The  addition  of  superheated  steam  may  result  in  the 
superheat  bemn  being  sufficient  to  supply  the  heat 
taken  by  the  cylinder  walls  and  thus  preventing  the 
condensation  and  making  available  the  area  kbcl. 
The  econxSmy  is  further  increased  as  the  steam  at  the 


FIG. 


end  of  expansion  has  less  moisture  in  it  and  thus  ab- 
stracts less  heat  from  the  cylinder  walls  during  release. 
That  is,  the  conduction  of  heat  through  a  vapor  occurs 
but  slowly,  while  water  in  contact  with  the  metal  will 
abstract  large  quantities  of  heat  during  evaporation. 
The  leakage  loss  is  also  less  with  superheated  steam. 
Loss  in  Efficiency  Due  to  Incomplete  Expansion. — 
If  steam  be  taken  throughout  the  entire  stroke  the 
indicator-card  is  represented  by  abed  (Fig.  92).  The 


240       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

drop  in  pressure  be  is  equivalent  to  cooling  at  constant 
volume  and  may  be  represented  on  the  7^-diagram 
by  the  curve  of  constant  volume  be.  If  the  same 
quantity  of  steam  be  taken  successively  into  larger 
cylinders,  so  that  an  increasing  degree  of  expansion 
is  obtained,  this  will  be  represented  by  be,  be',  be",  etc., 
in  both  diagrams.  The  areas  B,  C,  D  show  the  extra 
work  performed  per  pound  in  the  pi>-plane,  and  the 


FIG.  92. 

extra  heat  utilized  in  the  770-plane  respectively,  as  the 
expansion  progresses  from  initial  to  final  pressure. 

As  in  the  gas-engine,  so  in  the  steam-engine  it  seldom 
pays  to  carry  the  expansion  completely  down  to  back 
pressure,  because  the  slight  gain  from  c'  to  c"  is  more 
than  counterbalanced  by  the  increased  size,  cost,  and 
weight  of  the  engine,  friction,  and  radiation  losses,  etc. 

For  such  incomplete  expansion  the  expression  for  the 
efficiency  of  the  Rankine  cycle  is  found  as  follows: 


NON-CONDUCTING  STEAM-ENGINE. 


241 


=  (qc  +xcpc)  -  (q2  +xep2)  +xer2 
=  qc  +  xcrc  —  q2  —Axcuc(pc  —pz), 
Qi-Qt     -,      We  +  qc  -  q*  -Axcuc(pc  -  pa) 

'=^ 


1- 


Loss  in  Eflriciency  from  Use  of  Throttling  Governor. — 

The  throttling  governor  acts  by  wire  drawing  the  steam 
to  a  lower  pressure.  Less  steam  is  thus  taken  per 
stroke,  as  the  volume  is  increased  by  both  the  reduced 
pressure  and  the  increased  value  of  x.  A  series  of 
cards  for  dropping  pressure  is  shown  in  Fig.  93.  The 


FIG.  93. 

7V/>-diagram  shows  the  decreased  efficiency  per  pound 
of  steam  for  the  same  cases.  During  wire  drawing  ths 
heat  remains  the  same,  but  the  entropy  increases,  as 


242       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

the  process  is  irreversible.  The  heat  rejected  increases 
as  the  initial  pressure  drops,  so  that  of  the  total  heat 
brought  in  a  smaller  quantity  is  changed  into  work  and 
the  efficiency  of  the  plant  decreased. 

Pounds  of  Water  per  Horse-power-hour  for  the  Ran- 
kine  Cycle. — It  is  customary  to  quote  the  economy  of 
an  engine  in  pounds  of  water  per  horse-power  per  hour. 
In  time  one  acquires  a  rough  knowledge  of  the  varia- 
tions in  water  consumption  for  engines  of  different 
sizes  under  varying  conditions,  but  this  is  at  the  best 
a  crude  method  of  comparison  between  the  results  of 
various  engines  unless  one  possesses  a  standard  of 
reference  and  can  show  how  closely  the  actual  engines 
have  approached  the  ideal  conditions  in  each  case. 
The  Rankine  cycle  represents  the  thermal  history  of 
the  working  fluid  in  a  plant  containing  no  heat  losses. 
The  maximum  amount  of  work  is  thus  obtained  from 
each  pound  of  steam  and,  therefore,  the  steam  con- 
sumption of  the  Rankine  cycle  represents,  for  any 
given  set  of  conditions,  the  minimum  amount  which 
must  be  used  per  hour  to  develop  a  horse-power  of 
work. 

A  horse-power  is  equal  to  33,000  foot-pounds  of  work 
per  minute,  so  that  the  heat  equivalent  of  a  horse- 
power is  2545  B.T.U.  per  hour.  The  heat  utilized  per 
pound  in  developing  useful  work  is  equal  to  the  differ- 
ence of  the  total  heats  at  the  beginning  and  end  of  the 
isentropic  expansion.  The  pounds  of  steam  required 


NON-CONDUCTIXG    STEAM-ENGIXE.  243 

to  develop  one  horse-power-hour  of  work  are  therefore 

2545 

equal  to  jj TT  • 

tii—tiz 

The  table  on  page  244  shows  the  effect  of  varying 
initial  anil  final  pressures  upon  the  water  consumption 
of  the  ideal  engine  using  dry  steam. 

Pounds  of  Water  per  Kilowatt-hour  for  the  Rankine 

1000 

Cycle. — A  kilowatt  is  equal  to  -=^-  H.P.,  and  is  there- 
fore equivalent  to  44,240  foot-pounds  of  work  per  minute, 
which  is  equivalent  to  the  utilization  in  work  of  3412 
B.T.U.  per  hour.  The  pounds  of  steam  required  to 
develop  one  kilowatt-hour  of  electrical  energy  are 

3412 

therefore  equal  to  -77 fr- 

fli  —  fi2 

These  values  could  be  readily  tabulated  if  necessary 
but  they  may  also  be  obtained  by  multiplying  the 
corresponding  values  for  pounds  of  steam  per  horse- 
power-hour by  the  factor  1.341. 

Use  of  Entropy  Diagram  or  Tables.  —  The  labor 
involved  in  computing  the  ideal  water  consumption 
and  the  Rankine  efficiency  for  any  given  case  can  be 
greatly  reduced  by  using  either  a  temperature-entropy 
chart  containing  constant-heat  curves  or  Peabody's 
entropy  tables.  The  method  of  using  to  obtain  the 
water  consumption  is  evident  from  the  expression 

2545 
Hi- Ha' 


214        THE   TEMPERATURE-ENTROPY  DIAGRAM. 


POUNDS  OF  DRY  STEAM  PER  HORSE-POWER-HOUR  FOR 
RANKINE  CYCLE. 


Initial  Condition. 

Back  Pressures,  Pounds  Absolute. 

Pi 

h 

20.0 

14.7 

5 

4 

3 

2 

i 

5.99 

170 

1340 

150.6 

72.9 

42.0 

25.0 

7.51 

180 

94.3 

60.8 

42.3 

29.6 

20.2 

9.34 

190 

59.3 

44.4 

33.6 

25.3 

18.10 

11.53 

200 

44.8 

35.8 

28.5 

22.4 

16.52 

14.13 

210 

36.0 

30.0 

24.8 

20.0 

15.28 

14.70 

212 

34.7 

29.2 

24.2 

19.5 

15.08 

17.19 

220 

225^2 

30.2 

25.8 

21.9 

19.0 

14.20 

20.78 

230 

908  !  8 

99.0 

26.2 

22.9 

19.7 

16  .  63 

13.32 

25.0 

240 

156.2 

66.1 

23.2 

20.6 

18.04 

15.46 

12.55 

29.8 

250 

86.3 

49.2 

20.8 

18.74 

16.78 

14.40 

11.88 

35.4 

260 

59.9 

39.4 

18.9 

17.24 

15.48 

13.56 

11.28 

41.8 

270 

46.4 

33.3 

17.43 

15.99 

14.45 

12.80 

10.76 

49.2 

280 

38.5 

28.8 

16.19 

14.92 

13.61 

12.14 

10.30 

57.5 

290 

32.6 

25.5 

15.18 

14.11 

12.91 

11.55 

9.92 

67.0 

300 

28.4 

23.0 

14.29 

13.33 

12.24 

11.07 

9.57 

77.6 

310 

25.3 

21.0 

13.52 

12.66 

11.69 

10.61 

9.24 

89.6 

320 

22.9 

19.3 

12.84 

12.06 

11.20 

10.22 

8.90 

103.0 

330 

20.9 

17.93 

12.26 

11.57 

10.78 

9.85 

8.67 

117.9 

340 

19.4 

16.78 

11.75 

11.10 

10.40 

9.52 

8.43 

134.5 

350 

18.05 

15.83 

11.27 

10.68 

10.02 

9.24 

8.20 

152.9 

360 

16.96 

14.93 

10.87 

10.32 

9.71 

8.96 

7.97 

173.2 

370 

15.95 

14.19 

10.47 

10.00 

9.40 

8.72 

7.81 

195.5 

380 

15.18 

13.54 

10.16 

9.69 

9.15 

8.49 

7.63 

220.0 

390 

14.42 

12.99 

9.83 

9.40 

8.90 

8.29 

7.46 

246.9 

400 

13.80 

12.46 

9.57 

9.15 

8.68 

8.09 

7.32 

276.3 

410 

13.25 

12.00 

9.32 

8.93 

8.48 

7.92 

7.18 

308.5 

420 

12.74 

11.60 

9.08 

8.71 

8.29 

7.76 

7.05 

336.2 

428 

12.38 

11.30 

8.91 

8.55 

8.14 

7.64 

6.95 

422.4 

450 

11.4 

10.55 

8.43 

8.12 

7.76 

7.31 

6.69 

678.5 

500 

10.1 

9.46 

7.73 

7.52 

7.43 

6.85 

6.31 

956.1 

540 

9.58 

9.00 

7.52 

7.29 

7.01 

6.66 

6.18 

1212 

570 

9.37 

8.83 

7.45 

7.23 

6.97 

6.65 

6.175 

1516 

600 

9.40 

8.87 

7.54 

7.32 

7.06 

6.74 

6.28 

1867 

630 

9.71 

9.19 

7.82 

7.61 

7.35 

7.02 

6.54 

2137 

650 

10.2 

9.96 

8.23 

8.00 

7.73 

7.38 

6  .  SS 

2431 

670 

11.2 

10.6 

9.89 

8.76 

8.46 

8.07 

7.50 

2748 

690 

18.1 

16.7 

13.2 

12.7 

12.1 

11.3 

10.3 

2882 

698 

25.4 

23.2 

18.2 

17.4 

16.5 

15.4 

13.9 

A'OX-COXDUCTIXG   STEAM-ENGINE.  245 

To  obtain  the  Rankine  efficiency  we  have 


which  may  be  regrouped  and  written  in  the  form, 


in  which  all  the  quantities  may  be  read  directly  from 
the  chart  or  the  tables. 

Heat  Consumption  of  the  Rankine  Cycle  in  B.T.U. 
per  Horse-power-minute. — It  is  customary  to  rate  the 
economy  of  an  engine  in  terms  of  the  heat  units  required 
to  produce  one  indicated  horse-power  per  minute. 
As  a  standard  of  comparison  the  heat  consumption  of 
the  Rankine  cycle  using  dry  saturated  steam  is  given 
in  the  table  on  page  246. 

The  heat  equivalent  of  one  horse-power  is  42.42  B.T.U. 
per  minute.  The  heat  required  by  the  engine  per  minute 
may  be  determined  by  dividing  the  heat  utilized  by  the 

thermal  efficiency,  therefore, 

4242 
B.T.U.  per  I.H.P.  per  minute  =  ^±l£ 

Or,  again,  the  heat  supplied  to  the  engine  per  horse- 
power-hour equals  the  pounds  of  steam  per  horse-power- 
hour  multiplied  by  the  hc?at  absorbed  per  pound  of  water 

in  the  boiler^  therefore, 

Lbs.  steam  X  (Hi  —  g2) 
B.T.U.  per  I.H.P.  per  minute  =  —       — •=. 


246        THE  TEMPERATURE-ENTROPY   DIAGRAM. 


B.T.U.    PER   HORSE-POWER-MINUTE   FOR   RANKINE 
CYCLE  USING  DRY  STEAM. 


Initial  Conditions 

Back  Pressure,  Pounds  Absolute. 

Pi 

t, 

20.0 

14.7 

5 

4 

3 

2 

1 

5.9 

170 

22170 

2516 

1232 

720 

439.3 

7.5 

180 

.... 

1582 

1031 

724 

515 

359.5 

9.3 

190 

1000 

755 

578 

441.3 

323.4 

11.5 

200 

758 

611 

492.5 

392.5 

296.9 

14.13 

210 

611 

514 

429.3 

351.4 

274.3 

14.70 

212 

590 

500 

420.0 

342  .  6 

271.4 

17.  19 

220 

3652 

514 

444.4 

380.8 

335.0 

256.4 

20.78 

230 

14550 

1612 

448 

394  .  8 

344.5 

294.4 

241.4 

25.0 

240 

250S 

1079 

397 

356  .  6 

315.9 

274.8 

228.0 

29.8 

250 

1391 

807 

359 

325.6 

294.7 

262.6 

216.5 

35.4 

260 

969 

648 

330 

300.5 

272.9 

242.5 

206.3 

41.8 

270 

752 

549 

302 

279.4 

255.5 

229.5 

197.4 

49.2 

280 

619 

477 

281.3 

262.2 

241.3 

218.4 

189.5 

57.5 

290 

520 

423 

264.6 

247.7 

229.5 

208.3 

182.9 

67.0 

300 

466 

382 

249.6 

235.1 

218.1 

200.1 

176.8 

77.6 

310 

416 

349.5 

236.8 

223.9 

208.8 

192.3 

171.2 

89.6 

320 

377 

322.5 

225.6 

213.7 

200.7 

185.7 

165.3 

103.0 

330 

345 

300.5 

215.8 

205.5 

193.6 

179.3 

161.4 

117.9 

340 

321 

281  .  9 

207.3 

197.7 

187.4 

173.7 

157.1 

134.5 

350 

299 

266.8 

199.4 

190.5 

180.7 

169.0 

153.2 

152.9 

360 

282 

251.9 

192.6 

184.  5 

175.4 

164.2 

149.1 

173.2 

370 

266 

240.1 

186.0 

179.1 

170.0 

160.1 

146.5 

195.5 

380 

253 

229.4 

180.7 

173.8 

166.2 

156.1 

143.3 

220.0 

390 

241 

215.6 

175.2 

168.9 

161.6 

152.7 

140.4 

246.9 

400 

231 

211.9 

170.8 

164.6 

157.9 

149.6 

138.0 

276.3 

410 

222 

204.5 

166.5 

160.9 

154.4 

146.3 

135.6 

308.5 

420 

214 

197.9 

162.5 

157.0 

151.2 

143  .  5 

133.3 

336.2 

428 

208 

193.1 

159.7 

153.9 

14S.6 

141.6 

131.6 

422.4 

450 

193.7 

181.8 

152.3 

147.8 

142.8 

136.3 

127.4 

678.5 

500 

170.8 

162.1 

138.9 

136.3 

136.1 

126.0 

119.7 

956.1 

540 

158.8 

151.7 

132.9 

129.9 

126.5 

121.8 

115.4 

1212 

570 

151.8 

145.5 

128.8 

126.2    123.0 

119.1 

113.2 

1516 

600 

146.9 

141.1 

126.2 

123.7    120.6 

114.2    111.4 

1867 

630 

143  .  7 

138.6 

124.5 

122.2    119.4 

116.0|   110.6 

2137 

650 

143.2 

138.0 

124  .  5 

122.3 

119.5 

116.1 

110.9 

2431 

670 

145.0 

139.9 

126.4 

123.8 

112.3 

117.8 

112.6 

2748 

690 

198.6 

1S7.0 

159.0 

154.6 

149.5 

143  .  2 

134.2 

2882 

698 

203.4 

192.7 

165  .  S 

161.6 

156.6 

150.2 

141.1 

NON-CONDUCTING  STEAM-ENGINE.  247 

The  heat  consumption  may  therefore  be  determined 
from  either  of  the  preceding  tables,  the  results  are  as 
shown  on  page  246. 

B.T.U.  per  Kilowatt-minute. — The  heat  equivalent  of 
one  kilowatt  is  56.9  B.T.U.  per  minute.  The  heat  re- 
quired by  the  engine  per  minute  to  produce  one  kilo- 
watt when  working  upon  the  Rankine  cycle  is  found  by 
dividing  56.9  B.T.U.  by  the  thermal  efficiency.  Or 
the  same  value  may  be  obtained  by  multiplying  the 
corresponding  value  for  B.T.U.  per  horse-power-hour 
taken  from  the  table  on  page  246  by  the  factor  1.341. 


CHAPTER  XII. 
THE  MULTIPLE-FLUID   OR  WASTE-HEAT   ENGINE. 

IN  the  discussion  of  the  Rankine  cycle  it  was  shown 
how  the  efficiency  of  the  steam-engine  could  be  increased 
by  raising  the  temperature  of  the  source  of  heat  or  by 
decreasing  that  of  the  refrigerator.  Due  to  the  course 
of  the  pT-curve  a  practical  upper  limit  is  soon  reached 
in  the  use  of  saturated  steam  due  to  the  rapid  increase 
of  pressure  at  upper  temperatures,  so  that  recourse  has 
to  be  taken  to  superheated  steam.  Again,  in  reducing 
the  back  pressure  a  slight  drop  in  pressure  means 
a  large  drop  in  the  exhaust  temperature,  but  a  practical 
limit  is  soon  reached  beyond  which  it  does  not  pay  to 
carry  a  vacuum. 

Theoretically,  at  least,  the  efficiency  could  be  increased 
by  using  for  the  higher  temperatures  some  fluid  (A") 
having  a  smaller  vapor  pressure  than  saturated  steam, 
and  for  lower  temperatures  some  fluid  (Y)  having  a 
greater  vapor  pressure  than  saturated  steam  at  the 
same  temperature.  That  is  to  say,  the  first  fluid  X 
could  in  a  saturated  condition  be  heated  to  the  tempera- 
ture now  common  for  superheated  steam,  and  then  be 

248 


MULTIPLE-FLUID  OR  WASTE-HEAT  EXGL\E.     249 

allowed  to  expand  in  a  cylinder  down  to  some  lower 
temperature  at  which  the  pressure  of  saturated  steam 
would  not  be  excessive.  The  surface  condenser  for 
this  X-fluid  would  be  at  the  -same  time  the  boiler  for 
the  stealh.  After  the  steam  had  expanded  through 
two  or  three  cylinders,  it  in  turn  would  exhaust  into  a 
second  surface  condenser,  which  would  be  the  boiler 
for  the  next  fluid,  Y,  in  the  series.  Thus  the  working 
substance  in  each  case  is  condensed  at  the  temperature 
of  its  own  exhaust  and  fed  back  to  its  own  boiler  at 
this  temperature.  The  heat  of  vaporization  which  the 
first  fluid  rejects  must  warm  up  the  second  as  it  is  fed 
into  the  boiler  and  then  vaporize  part  of  it,  so  that 

heat  rejected  =  heat  received, 

OT  ^exhaust  -  (5  +  *r)boiler  "^condenser" 

For  the  sake  of  simplicity  suppose  at  first  that  two 
such  fluids  X  and  Y  as  described  could  be  found,  and 
further  that  their  liquid  and  saturated -vapor  lines  coin- 
cided in  the  T<£-plane  with  those  for  water,  but  that 
the  Tp-curves  are  entirely  different. 

Suppose,  further,  that  pt  and  p2  (Fig.  94)  represent 
respectively  the  highest  pressure  which  it  is  convenient 
to  use,  and  the  lowest  pressure  which  can  be  obtained 
in  a  vacuum.  The  range  of  temperatures  when  satu- 
rated steam  alone  is  used  is  limited  between  ^  and  t2. 
If,  however,  the  fluid  X  is  first  used  the  upper  tempera- 
ture can  be  raised  to  tx  for  the  same  maximum  pres- 


250        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

sure  Pi.  Suppose  in  this  problem  that  the  substance  Y 
does  not  solidify  at  32°  F.,  its  liquid  line  would  extend 
to  the  left  of  the  arbitrary  zero  for  the  entropy  of  water, 
and  so  the  expansion  of  this  fluid  down  to  p2  would 
drop  the  lower  temperature  from  t2  down  to  ty. 


The  liquid  A"  is  fed  into  its  boiler  at  the  temperature  tl} 
and  is  warmed  along  ab,  receiving  the  heat  qb~qa, 
equal  to  area  ablm;  it  is  then  vaporized  at  the  pres- 
sure Pi  and  receives  the  heat  rx  equal  to  bcnl.  Its  ideal 
cycle  is  now  completed  by  adiabatic  expansion  cd 
down  to  some  pressure  px  (in  Fig.  94  px  coincides 
with  p2],  corresponding  to  temperature  tlt  on  the  X- 


MULTIPLE-FLUID  OR  WASTE-HEAT  ENGINE.     251 

curve,  and  condensation  along  da.  The  condensed 
fluid  is  at  the  proper  temperature  to  be  returned  to 
its  boiler. 

The  heat  rejected  along  da,  viz.,  xdrd,  must  warm 
up  the  water  fed  into  the  X-condenser  at  temperature 
t3  up  to  ^  and  then  vaporize  part  of  it  at  the  upper 
temperature.  Assuming  no  heat  lost, 

area  m'gaen'  =  area  madn. 

The  steam  describes  the  ideal  cycle  gaef,  rejecting  in 
turn  the  heat  under  fg,  equal  to  xtrf  at  some  pressure 
p3  corresponding  to  temperature  t3. 

In  the  steam-condenser  the  fluid  Y  is  first  warmed 
up  from  the  temperature  tY}  corresponding  to  p2,  to  t3 
and  then  enough  vaporized  to  make 

area  m"kghn"  =  area  m'gfn'. 

From  the  diagram  the  following  conclusions  can  be 
drawn : 

Heat  received  from  fuel  equals  mabcn. 
Heat  utilized  by  X-engine  equals  abed. 

-rffi  •  c  v        • 

Efficiency  of  X-engme,  ^ 

Heat  rejected  by  X-engine  =  area  wadn  =  area  m'gaen' 
=  heat  absorbed  by  steam-engine. 
Heat  utilized  by  steam-engine  equals  gaef. 


252        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


Efficiency  of  steam-engine  fjst  =  -~  -  -,. 

Efficiency  of  X-  and  steam-engines  combined  equals 


abed  +gaef 
mabcn 


Heat  received  by  F-engine  =  heat  rejected  by  steam- 
engine  =  m"kghn"  •=  m'gfn'. 

Heat  utilized  by  F-engine  =  kghi. 
Heat  rejected  by  Y-engine  =  w"A;in". 

Efficiency  of  F-engine         =     ////. 


Efficiency  of  all  three  engines  together, 
abed  +gaef  +kghi 

' 


The  heat  rejected  has  been  reduced  from 
madn  to  mffkthfednmrf. 

The  great  gain  in  efficiency  shown  in  this  assumed 
case  is  deceptive.  The  exhaust  temperature  has  been 
taken  far  below  freezing.  This  could  not  be  done 
unless  some  cooling  mixture  could  be  employed  in  order 
to  condense  the  exhaust  fluid  Y. 

This  would  be  expensive  and  probably  represent  as 
great  an  expenditure  of  work  as  the  increased  gain 


MULTIPLE-FLUID  OR  WASTE-HEAT  ENGINE.     253 

recorded  by  the  F-piston,  possibly  more.  In  prac- 
tice the  lowest  temperature  tY  will  be  governed  by 
that  of  the  cheapest  available  condensing  substance, 
i.e.,  the  temperature  of  the  cooling  water  at  the  power- 
station  . 

The  upper  temperature  tx  will  be  governed  by  the 
materials  used  in  construction. 

Since  xexrex  =  qB—  qc  +xBr  B,  it  follows  that  xB<xex; 
that  is,  the  value  of  x  gradually  grows  smaller  for 
succeeding  fluids,  so  that  initial  condensation  must 
be  increasing  in  the  successive  cylinders.  The  gain 
which  accrues  from  the  use  of  superheated  steam 
might  for  similar  reasons  be  expected  from  the  use 
of  the  superheated  vapors  of  the  X-  and  F-fluids.  It 
thus  would  undoubtedly  pay  to  superheat  each  vapor 
as  it  leaves  its  respective  boiler  an  amount  sufficient 
to  overcome  the  initial  condensation.  This  might  be 
effected  by  the  use  of  a  separately  fired  superheater 
suitably  situated,  or  perhaps  more  economically  still 
the  hot  flue  gases  from  the  first  boiler  furnace  might 
be  made  to  pass  successively  through  all  the  super- 
heaters and  thus  the  total  economy  increased  two 
ways  at  once. 

Several  different  multiple-fluid  engines  have  been 
proposed,  usually  of  the  binary  type.  When  it  comes 
to  a  discussion  of  any  particular  combination  the 
actual  T0-diagram  must  be  drawn,  and  the  liquid- 
and  saturated -vapor  lines  will  no  longer  be  super- 


254        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

imposed  as  was  assumed  in  the  ideal  case  just  discussed. 
However,  the  principles  already  enunciated  will  make 
the  application  clear. 

Any  fluid  having  a  low  boiling-point,  such  as  ammonia, 
chloroform,  sulphur  dioxide,  ether,  carbon  disulphide, 
etc.,  is  available  for  such  work.  All  such  volatile 
fluids  possess  latent  heat  of  vaporization  of  small 
magnitude,  and  the  smaller  this  is,  the  more  volatile 
the  substance  and  the  greater  its  specific  pressure  at  a 
given  temperature.  This  leads  to  the  practically 
important  fact  that  to  perform  a  given  amount  of 
work  a  greater  quantity  of  the  volatile  substance 
must  be  supplied,  the  amount  necessary  increasing 
with  the  diminution  of  the  latent  heat  of  vaporization. 

Du  Trembley  used  a  steam-ether  engine.  Ether 
superheats  during  adiabatic  expansion  and  thus  seems 
especially  adapted  to  such  work,  as  this  would  tend  to 
prevent  the  excessive  cooling  of  the  cylinder  during 
exhaust  and  thus  do  away  with  the  losses  incident 
to  initial  condensation.  Fig.  19,  although  not  drawn 
to  accurate  scale,  gives  an  approximate  idea  of  the  rela- 
tive values  of  the  latent  heat  of  vaporization.  In  round 
numbers  the  entropy  as  liquid  and  as  vapor  compares 
with  water  as  follows: 


32°  F..  0  =  0,          0+^ 
Water 

248°  F..  0  =  0.  36.5,  0+^ 


MULTIPLE-FLUID  OR  WASTE-HEAT  ENGINE.    255 


Ether 


32°  F..  0  =  0,          0+^  =  0.34; 


248°  F.  .0-0.205,  04-    =  0.385; 


and  as  the  latent  heat  is  about  one  sixth  that  of  water 
it  follows  that  about  six  pounds  of  ether  will  be  re- 
quired to  cool  each  pound  of  steam,  so  that  a  com- 
bined TV-diagram  might  be  drawn  for  one  pound  of 
water  and  six  of  ether. 

Perhaps  the  most  accurate  and  elaborate  series  of 
experiments  on  any  binary  engine  was  made  by  Prof. 
Josse  of  Berlin.  He  used  sulphur  dioxide  for  the 
secondary  fluid.  The  best  results  obtained  were 
11.2  pounds  of  steam  per  horse-power  for  the  steam- 
engine  alone  and  the  equivalent  of  but  8.36  pounds  per 
horse-power  per  hour  using  the  combined  engine.  In 
the  test  the  "waste-heat"  engine  added  34.2  per  cent, 
to  the  power  obtained  from  the  primary  engine. 

The  steam  had  a  pressure  of  171  pounds  absolute 
and  was  superheated  to  558°  F.  The  back  pressure  on 
the  low-pressure  cylinder  of  the  steam-engine  was  about 
2.9  pounds  absolute,  corresponding  to  140°  F.  The 
S02  cylinder  received  vapor  under  pressure  of  143 
pounds  and  exhausted  at  48.2  pounds  absolute,  cor- 
responding to  a  temperature  of  67.8°  F. 

In  round  numbers  the  latent  heat  of  S02  is  about 
one  seventh  that  of  water,  so  that  it  would  require  the 


256        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

vaporization  of  from  6  to  7  pounds  of  S02  to  condense 
1  pound  of  steam. 

Fig.  95  shows  the  ideal  cycle  for  a  binary  engine  of 
this   type   working   between   the  pressures   and   tem- 


FIG.  95. 

peratures  realized  by  Prof.  Josse  in  the  test  at  Charlot- 
tenburg. 

The  steam-engine  was  triple  expansion,  and  the 
ideal  cards  for  such  an  engine  are  shown  combined  at 
abcde.  The  heat  exhausted  to  the  steam-condenser, 
or  S02  boiler,  xr,  equals  aerm.  Of  this  quantity  an 


MULTIPLE-FLUID  OR  WASTE-HEAT  ENGINE    257 

amount  hifg  is  saved  by  the  S02  cylinder  theoretically. 
Actually  there  was  a  drop  in  temperature  of  about  6°  F. 
between  the  low-pressure  steam-cylinder  and  the  S02 
cylinder,  so  that  the  heat  area  iff'i'  was  either  totally 
lost  or  partially  reduced  to  lower  efficiency  by  wire- 
drawing. 

Dr.  Schreber,  in  Die  Theorie  der  Mehrstoffmaschinen, 
proposes  the  following  combination  of  fluids: 

Substance.  Temperature  range. 

Aniline 590°  F.-3740  F. 

Water 374°  F.-1760  F. 

jEthylamine 176°  F.-860  F. 

Josse,  in  Neuere  Warmekraftmaschinen  (Berlin,  1905), 
publishes  reports  of  tests  on  several  H20-S02  engines. 
The  S02-cylinders  developed  from  450  H.P.  to  150  H.P., 
and  had  been  in  operation  from  one  to  two  years.  The 
results  show  that  the  machines  have  come  up  to  the 
guarantee  of  the  manufacturers.  Josse  states  that  in 
view  of  the  high  initial  cost  such  machines  should 
probably  not  be  installed  unless  the  plant  is  to  run  a 
large  number  of  hours  daily,  and  unless  a  sufficient 
amount  of  cold  cooling  water  is  available. 


CHAPTER  XIII. 

THE  TEMPERATURE-ENTROPY   DIAGRAM  OF  THE 
ACTUAL  STEAM-ENGINE  CYCLE. 

THE  Rankine  cycle  is  based  upon  the  following  as- 
sumptions : 

(1)  Non-conducting  cylinder  walls  and  piston; 

(2)  Isentropic  expansion  to  the  back  pressure; 

(3)  Instantaneous  action  of  the  valves; 

(4)  No  leakage  by  the  piston  and  the  valves. 

From  the  first  two  conditions  it  follows  that  the  size 
of  the  clearance  space  is  immaterial. 

Referring  to  the  actual  steam-engine,  we  find  that 
the  conductivity  of  the  metal  produces  initial  conden- 
sation and  reevaporation  losses,  and  that  the  expansion 
can  be  carried  to  back  pressure  only  by  reducing  the 
efficiency.  The  size  of  the  clearance  must  therefore  be 
considered,  because  the  cycle  of  the  clearance  steam  will 
affect  the  economy.  The  valves  are  not  instantaneous 
in  action,  and  leakage  always  occurs  by  both  piston  and 
valves.  The  Rankine  cycle  is  thus  unattainable  in 
practice  and  is  but  an  ideal  which  the  actual  engine 

strives  to  approximate. 

258 


ACTUAL  STEAM-ENGINE  CYCLE.  259 

The  amount  of  condensation  and  reevaporation  is 
the  result  of  so  many  factors,  that  to  determine  the 
influence  of  each  by  the  ordinary  methods  of  compari- 
son would  require  too  much  time  and  money.  Hence 
to  aid  in  evolving  a  theory  of  the  steam-engine  which 
shall  account  for  all  heat  losses  snd  interchanges,  some 
convenient  form  of  analysis  must  be  adopted  by  which 
the  losses  for  any  single  test  may  be  investigated. 

Hirn's  analysis  makes  possible  the  determination  of 
the  net  heat  changes  occurring  between  admission  and 
cut-off,  cut-off  and  release,  release' and  compression,  and 
compression  and  admission,  but  does  not  give  informa- 
tion as  to  the  actual  direction  of  heat-transference  at 
any  moment.  Fortunately,  the  T^-diagram  offers  a 
graphical  solution  equivalent  to  that  of  Hirn's  analysis, 
and  also  makes  clear  the  direction  in  which  the  inter- 
change of  heat  is  occurring  at  any  point.  Before 
a  T (^-projection  of  an  indicator-card  can  be  made, 
it  will  be  necessary  to  discuss  at  length  the  different 
lines  of  the  card  in  order  to  determine  exactly  what 
each  represents. 

The  Admission  Line  of  the  Indicator-card. — During 
admission  the  steam  is  not  at  a  uniform  temperature 
and  pressure.  Part  is  still  in  the  steam-pipe  under 
boiler  pressure,  part  has  passed  through  the  valve-chest 
and  steam-ports,  and  has  already  entered  the  cylinder, 
and  still  a  third  portion  is  in  the  process  of  transition. 
In  general,  the  surrounding  metal  is  colder  than  the 


260        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

steam,  so  that  a  continual  loss  of  heat  is  experienced 
resulting  in  condensation  and  decrease  of  volume  and 
entropy.  To  this  is  added  the  further  effect  of  wire- 
drawing, due  to  too  small  steam  passages  and  to  the 
throttling  effect  when  the  valve  is  opening  and  closing, 
producing  a  drop  of  pressure  and  increase  of  both  spe- 
cific volume  and  entropy.  It  is  probable  that  each 
particle  of  steam  follows  its  own  path  in  passing  from 
the  steam-pipe  into  the  cylinder  up  to  the  point  of  cut- 
off. Thus  the  admission  line  of  the  indicator-card  is 
not  the  /^-history  of  the  entire  quantity  of  steam  nor 
of  any  particular  part  of  it,  and  is  only  a  record  of  the 
pressure  exerted  from  moment  to  moment  by  the  vary- 
ing quantity  of  steam  confined  in  the  cylinder.  Hence 
in  projecting  the  admission-curve  into  the  T<£-plane  it 
must  be  remembered  that  the  projection  does  not  rep- 
resent the  T^-history  of  any  portion  of  the  steam,  but 
is  simply  a  reproduction  of  each  individual  point  of  the 
pv-curve. 

Let  a'c'  (Fig.  96)  represent  the  T^-projection  of  the 
admission  line  of  an  indicator-card,  while  b  represents 
the  state  point  of  the  steam  in  the  steam-pipe.  But 
for  the  various  losses  the  admission  line  would  have 
been  ab,  which  represents  the  actual  path  followed  by 
the  steam  in  the  boiler.  If,  for  a  moment,  we  consider 
the  admission  to  represent  a  reversible  process,  the 
area  under  a'c'  will  represent  the  heat  received  during 
this  process.  Hence  the  area  abb^c'a'  represents  the 


ACTUAL  STEAM-ENGINE  CYCLE. 


261 


difference  between  the  heat  contained  per  pound  of 
steam  in  the  boiler,  and  the  amount  realized  per  pound 
in  the  cylinder,  or  the  losses  due  to  initial  condensation 
and  wire-drawing.  The  former  would  simply  result  in 
the  condensation  of  part  of  the  steam,  thus  causing  the 
value  of  x  to  diminish  from  b  to  c,  or  possibly  to  some 
point  slightly  to  the  left  of  c;  the  latter  would  cause  a 


FIG.  96. 

drop  in  pressure  and  increase  of  entropy,  moving  the 
state  point  to  c' '.  Due  to  the  impossibility  of  distin- 
guishing accurately  between  these  two  opposing  fac- 
tors, one  tending  to  decrease,  the  other  to  increase  the 
entropy,  the  area  cbb^  is  taken  to  represent  the  loss 
due  to  initial  condensation,  and  the  area  ace' a'  to  repre- 
sent the  loss  or  reduction  in  efficiency  due  to  friction 
and  wire-drawing. 


202         THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The    Expansion  Line   of  the    Indicator-card. — If  we 

assume  that  the  leakage  by  piston  and  valves  is  negli- 
gible during  expansion,  the  expansion-curve  between 
cut-off  and  release  represents  the  continuous  pv-history 
of  the  entire  quantity  of  steam  contained  in  the  cylin- 
der; that  is,  of  the  cylinder  feed  plus  the  clearance 
steam.  The  temperature  of  the  steam  throughout  the 
cylinder  is  not  uniform,  as  heat-conduction  is  occurring 
between  the  steam  and  the  metal,  so  that  the  indicator 
records  but  the  average  pressure  due  to  these  variable 
temperatures.  Hence  the  ^-projection  will  give  but 
average  values  of  the  T0-changes  during  expansion. 

Since  there  is  no  appreciable  friction  of  the  steam 
against  the  metal,  as  during  admission,  it  follows  that 
neglecting  the  inequalities  in  the  temperature  of  the 
steam,  there  is  no  reduction  of  the  heat  efficiency  of 
the  steam  due  to  internal  irreversible  processes,  and 
thus  any  increase  or  decrease  of  the  entropy  of  the 
steam  must  result  from  heat-transferences  between  the 
steam  and  the  surrounding  metal.  If  adiabatic,  the 
curve  would  here  be  isentropic,  but  as  the  steam  is  at 
first  hotter  than  the  cylinder  walls,  the  flow  of  heat  is 
from  steam  to  metal,  thus  causing  an  increase  in  the 
entropy  of  the  metal  and  a  decrease  in  that  of  the 
steam. 

The  expansion  line  thus  assumes  at  the  start  some 
such  form  as  c'd,  becoming  steeper  as  the  temperature 
drops,  and  just  at  the  moment  the  temperatures  of  the 


ACTUAL  STEAM-ENGINE  CYCLE.  263 

steam  and  the  walls  are  the  same  it  becomes  isentropic. 
From  this  point,  d,  on  to  release  at  e,  the  heat  transfer 
is  from  metal  to  steam,  so  that  the  entropy  of  the  latter 
now  increases  and  the  curve  slants  to  the  right. 

The  ideal  engine,  supplied  with  steam  of  condition  c', 
would  expand  isentropically  along  c'c^  to  the  back  pres- 
sure at  h.  Hence  the  area  c'dd^,  under  the  first  part 
of  the  expansion-curve,  represents  the  loss  of  heat  due 
to  conduction.  Again,  the  ideal  engine,  supplied  with 
steam  of  condition  d,  would  expand  isentropically  to  k. 
The  area  under  de,  for  the  actual  engine,  thus  repre- 
sents a  gain  due  to  the  heat  returned  by  the  walls.  It 
should  be  noted  that  the  heat  thus  regained  is  restored 
at  a  lower  temperature  than  that  at  which  it  was  lost, 
and  hence  at  a  lower  efficiency. 

The  Exhaust  Line  of  the  Indicator-card. — Let  us  con- 
sider first  the  case  where  the  expansion  is  carried  down 
to  back  pressure.  The  ideal  engine,  supplied  with 
steam  of  quality  e,  would  expand  along  ee1  down  to  the 
pressure  in  the  condenser  and  then  condense  along  mg. 
The  actual  engine,  due  to  the  resistance  of  the  exhaust 
ports,  etc.,  would  expand  to  some  pressure,  as  I,  greater 
than  that  in  the  condenser  and  would  then  exhaust 
along  lg'.  The  area  Ig'gm  would  thus  represent  the 
loss  of  heat  due  to  throttling  during  exhaust. 

If  the  release-valve  opens  at  e  before  back  pressure 
is  reached,  the  phenomena  are  as  follows:  As  the  valve 
starts  to  open  steam  begins  to  escape  and  is  throttled 


264        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

down  to  the  condenser  pressure ;  as  the  valve  continues 
to  open  the  escape  becomes  more  rapid  until  the  back 
pressure  is  established.  Then  on  the  return  stroke  more 
steam  is  forced  out  against  the  back  pressure,  and  near 
the  end,  as  the  exhaust -valve  begins  to  close,  there  is 
a  slight  rise  in  pressure  and  a  small  quantity  escapes, 
suffering  reduction  in  efficiency  by  throttling.  As  the 
valve  closes,  all  the  cylinder  feed  has  escaped,  and  only 
the  clearance  steam  remains.  It  is  necessary  to  note 
that  the  exhaust  line  of  the  card  records  the  pressure 
of  the  steam  still  in  the  cylinder  at  any  moment  and 
gives  no  information  whatever  as  to  its  condition,  or 
of  the  condition  of  that  portion  already  exhausted. 
Thus  part  of  the  steam  has  already  reached  the  con- 
denser (or  in  the  case  of  a  multiple-expansion  engine 
the  following  cylinder  or  intermediate  receiver),  and 
has  already  parted  with  some  of  its  heat,  while  that 
still  in  the  cylinder,  being  at  a  lower  temperature  than 
the  cylinder  walls,  is  receiving  heat  and  losing  its  mois- 
ture and  may  sometimes  at  compression  have  become 
even  superheated.  The  last  part  of  the  exhaust-steam 
will  necessarily  have  to  retrace  part  of  this  thermody- 
namic  process  on  reaching  the  condenser,  or  upon  min- 
gling with  the  rest  of  the  steam  in  the  following  receiver 
or  cylinder. 

The  exhaust  line  of  the  card  does  not  represent  the 
p-u-history  of  any  definite  quantity  of  steam,  but  is  sim- 
ply a  pressure  record  of  continually  varying  quantities 


ACTUAL  STEAM-ENGINE  CYCLE.  263 

confined  in  a  constantly  diminishing  volume.  It  does, 
however,  represent  the  amount  of  work  required  to 
discharge  the  steam,  and  in  that  sense  the  area  under 
its  TV-projection  will  represent  the  total  heat  dis- 
charged. 

In  the  case  of  the  ideal  engine,  the  exhaust  line,  efg, 
divides  into  two  parts,  ef  and  fg,  equivalent  to  decrease 
of  pressure  at  constant  volume  and  to  decrease  of  vol- 
ume at  constant  pressure  respectively.  The  heat  re- 
jected is  represented  by  the  total  area  under  efg,  and 
exceeds  that  rejected  after  complete  expansion  to  the 
back  pressure  by  efm,  which  thus  represents  the  extra 
loss  incurred  by  incomplete  expansion.  The  exhaust 
line  for  the  actual  indicator-card  will  be  some  such 
curve  as  efg',  where  the  area  ef'l  shows  the  loss  due  to 
incomplete  expansion,  and  Ig'gm  the  loss  due  to  throt- 
tling, friction,  etc. 

The  Compression  Line  of  the  Indicator-card. —  The 
compression-curve,  from  the  closing  of  the  exhaust- 
valve  up  to  the  moment  of  admission,  gives  the  pv- 
history  of  the  clearance  steam,  and,  if  no  leakage  is 
assumed,  the  T ^-projection  will  thus  be  the  actual 
T^-history  of  a  definite  quantity  of  steam.  As  the 
pressure  increases  the  curve  deviates  more  and  more 
rapidly  from  the  adiabatic,  due  to  the  increasing  effect 
of  conduction  losses,  and  on  some  cards  may  become 
nearly  isothermal.  In  such  cases  it  is  probable  that 
the  assumption  of  dry  steam  at  compression  is  incor- 


266        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

rect,  the  presence  of  moisture  helping  to  explain  the 
rapid  loss  of  heat. 

During  the  interval  between  the  opening  of  the  ad- 
mission-valve and  the  attainment  of  initial  pressure 
the  time  interval  is  so  small  that  probably  the  assump- 
tion of  adiabatic  compression  of  the  clearance  steam 
would  not  be  greatly  wrong.  The  gain  in  heat  thus 
incurred  must  be  at  once  lost  by  condensation  during 
the  first  part  of  the  admission,  but  it  is  impossible  to 
determine  the  history  of  this  change. 

To  obtain  the  ^-projection  of  the  compression- 
curve,  the  saturation-curve  for  the  weight  of  clearance 
steam  should  be  drawn  through  the  point  of  compres- 
sion (assuming  dry  steam  at  compression)  and  the 
projection  performed  as  previously  described.  The 
curve  will  assume  some  such  form  as  pq,  which  may 
or  may  not,  according  to  circumstances,  have  its  course 
partially  or  wholly  in  the  saturated  or  superheated 
regions.  In  any  case  the  area  under  the  curve,  when 
reduced  to  the  proper  ratio,  shows  the  heat  lost  to  the 
walls  during  compression,  and,  if  the  horizontal  line  of 
the  indicator-card  is  established  at  q,  gives  a  general 
idea  of  the  temperature  of  the  cylinder  at  the  moment 
of  admission,  and  hence  a  measure  of  the  heat  neces- 
sary to  bring  the  cylinder  up  to  the  temperature  of 
the  entering  steam.  Except  for  this  one  feature,  the 
cycle  of  the  clearance  steam  is  unimportant,  as  all 
the  losses  occasioned  by  it  will  be  manifested  by  the 


ACTUAL  STEAM-ENGINE  CYCLE.  267 

difference  between  the  cycle  for  an  ideal  engine, 
working  between  the  given  pressures  in  the  steam 
pipe  and  condenser,  and  the  T<}>-p\ot  of  the  actual 
card. 

The  Indicator-card. — Considered  as  a  whole,  the  indi- 
cator-card furnishes  the  following  information.  The 
admission  line  and  the  exhaust  line  simply  represent 
the  pressure  of  part  of  the  steam,  but  do  not  give  any 
information  regarding  the  specific  volume.  On  the 
other  hand,  both  the  expansion  and  the  compression 
lines  give  the  history  of  definite  quantities  of  steam. 
Thus  both  the  expansion  and  compression  of  the 
clearance  steam  are  recorded,  while  only  the  expan- 
sion of  the  cylinder  feed  appears,  the  compression  of 
the  latter  occurring  in  the  boiler.  The  entire  card,  if 
plotted  directly,  can  at  the  best  be  considered  only  as 
the  heat  equivalent  of  the  work  done,  but  not  as  the 
!T<£-history  of  any  closed  cycle.  Thus,  for  example^ 
while  the  projection  of  the  exhaust  line  gives  some  such 
curve  as  ef'g',  which  on  the  !T<£-chart  indicates  con- 
densation, it  is  probable  that  the  value  of  x  of  the  con- 
fined steam  is  actually  increasing. 

The  difficulties  involved  in  the  proper  interpretation 
of  the  irreversible  portions  of  the  indicator-card  have 
led  different  investigators  to  make  certain  assumptions 
as  to  the  influence  of  the  clearance  eteam  and  as  to  the 
possibility  of  replacing  the  actual  curves  by  equivalent 
reversible  processes. 


268        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  Clearance  Steam  Considered  as  an  Elastic  Cushion. 
— During  expansion  the  clearance  steam  follows  the 
same  laws  and  variations  as  the  cylinder  feed,  and  this 
hi  general  is  not  the  reverse  of  its  history  during  com- 
pression. Thus  the  cycle  of  the  clearance  steam,  if  it 
could  be  drawn,  would  enclose  an  area  representing 
either  positive  or  negative  work.  This  cycle  would 
then  be  of  especial  interest  in  determining  losses  ex- 
perienced by  the  clearance  steam,  but  as  these  losses 
must  eventually  be  charged  against  the  entering  steam, 
the  total  effect  upon  the  efficiency  would  be  the  same 
if  the  clearance  steam  were  considered  as  an  isolated 
elastic  cushion  expanded  and  compressed  along  the 
same  adiabatic.  If,  then,  an  adiabatic  is  drawn  through 
the  point  of  compression  on  the  indicator-card,  the 
horizontal  distance  from  any  point  on  this  adiabatic  to 
the  corresponding  point  on  the  indicator-card  shows 
the  volume  which  the  cylinder  feed  would  occupy 
under  the  above  assumptions.  Taking  this  adiabatic 
as  the  line  of  zero  volume,  a  diagram  can  thus  be  con- 
structed which  shows  only  the  variations  of  the  cylinder 
feed.  It  is  then  only  necessary  to  draw  on  the  satura- 
tion-curve for  the  weight  of  steam  fed  to  the  cylinder 
per  stroke  and  the  card  can  be  at  once  projected  into 
the  T$-plane.  This  is,  in  its  essential  points,  the 
method  adopted  by  Prof.  Reeve  in  his  book  on  the 
Thermodynamics  of  Heat  Engines,  although  he  changes 
volumes  so  that  the  reconstructed  card  represents 


ACTUAL  STEAM-ENGINE  CYCLE.  269 

the  pr-history  of  one  pound  of  cylinder  feed  instead 
of  the  actual  weight  in  the  cylinder.  To  project  his 
.reconstructed  card  into  the  T^-plane,  it  is  only  nec- 
essary to  draw  the  saturation-curve  for  one  pound  of 
steam  instead  of  that  for  the  pounds  fed  per  stroke. 
Inasmuch  as  the  T ^-projection  will  be  the  same  in 
either  case  it  seems  somewhat  simpler  to  adopt  the 
first  method,  viz.,  to  construct  the  saturation-curve  for 
the  number  of  pounds  in  the  cylinder  rather  than  to 
redraw  the  diagram  to  correspond  to  one  pound  of 
cylinder  feed. 

This  method  undoubtedly  makes  possible  a  deter- 
mination of  the  general  magnitude  and  character  of  the 
various  heat  interchanges,  but  is  open  to  the  following 
objections. 

The  compression  line  of  the  card  refers  to  the  clear- 
ance steam  alone,  so  that  the  deviations  from  the  adia- 
batic  thus  obtained  refer  to  itself  and  not  to  the  cylin- 
der feed.  Furthermore,  the  reconstructed  curve  may 
actually  pass  to  the  left  of  the  water  line,  assuming 
imaginary  values  on  the  T<£-plane,  and  thus  give  a 
wrong  conception  of  the  condition  of  this  steam,  which, 
instead  of  being  wet,  is  usually  dry  or  superheated,  and 
lies  to  the  right  of  the  expansion  line.  Again,  the  ex- 
pansion line  no  longer  represents  the  actual  pv-  or 
T<£-history  of  the  steam,  but  an  imaginary  history  which 
the  cylinder  feed  might  have  if  the  clearance  steam 
expanded  adiabatically.  The  entire  card  thus  becomes 


270        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

to  a  certain  extent  imaginary,  and  in  so  far  is  unde- 
sirable. 

The  Indicator-card  Considered  as  a  Reversible  Cycle. 
— The  area  of  the  card  gives  the  heat  changed  into  work, 
and  this  same  result  may  be  attained  by  assuming  the 
clearance  steam  and  cylinder  feed  to  remain  in  the 
cylinder,  being  heated  and  cooled  by  external  means 
and  thus  caused  to  expand  and  contract  along  a  rever- 
sible cycle  coincident  in  shape  with  the  actual  card. 
The  original  card  may  thus  be  projected  into  the  T(f>- 
plane  as  soon  as  the  saturation-curve  for  the  total 
weight  of  steam  has  been  drawn  on  it.  The  expansion 
line  represents  the  actual  history  of  the  substance,  but 
the  compression  line  is  entirely  imaginary. 

This  method  was  adopted  by  Prof.  Boulvin  in  his 
book,  The  Entropy  Diagram,  from  which  the  following 
two  illustrations  are  copied,  with  but  slight  alterations. 
ABCDEF,  in  Fig.  97,  represents  the  ^-projection  of 
a  certain  indicator-card.  The  line  DE  represents  the 
actual  expansion  history  of  the  total  steam;  the  other 
lines  give  more  or  less  imaginary  values.  The  small 
diagram  at  the  right  is  used  to  interpret  the  compres- 
sion line  AB.  If  W  and  w  represent  the  pounds  of 
cylinder  feed  and  clearance  steam  respectively  per 
revolution,  the  large  diagram  represents  the  cycle  of 
W  +w  pounds  of  the  mixture,  while  the  small  diagram 
represents  the  entropy  of  w  pounds.  Assuming  dry 
steam  at  compression,  the  heat  rejected  by  w  pounds 


ACTUAL  STEAM-ENGINE  CYCLE. 


271 


during  compression  is  shown  by  area  abb^.  The 
entropy  a0a  =  A0A  is  that  due  to  the  vaporization  of 
w  pounds  of  water,  so  that  AG  must  represent  that  for 
the  W  pounds  of  cylinder  feed.  Through  A  and  G  it 
is  then  possible  to  draw  the  water  line  AL  and  the 
dry-steam  line  GX  for  W  pounds.  The  cycle  for  the 


;<*, 


FIG.  97. 

Rankine  engine  using  W  pounds  per  revolution  is  rep- 
resented by  ALXH,  so  that  the  efficiency  of  the  actual 
engine  as  compared  with  that  of  the  ideal  engine  is 

ABCDEF 
ALXH  ' 

In  Fig.  98,  let  L'  and  L  be  the  liquid  lines  of  W  and 
W  +w  pounds  respectively,  and  let  L"  be  drawn  through 


272        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


the  point  of  compression  a  parallel  to  L'  ;  and  let  S  and 
Sf  be  the  dry-steam  lines  corresponding  to  the  water 
lines  L  and  L"  '  .  As  is  evident  from  Fig.  97,  the  hori- 
zontal distance  between  the  liquid  line  A0C0  and  the 
compression  line  AB  is  equal  to  the  entropy  of  vapori- 
zation of  the  clearance  steam  (for  example,  A0A=atfi} 
BoB  =  b0b}.  In  Fig.  98  the  horizontal  distance  between 
L'  and  L  represents  the  entropy  for  W+w  —  W  or  w 


FIG.  98. 


pounds  of  water.  Therefore  the  horizontal  distance 
from  L'  to  the  compression-curve  db  represents  the 
total  entropy  of  the  clearance  steam.  a'U  is  thus  the 
curve  of  zero  entropy  for  the  clearance  steam,  and  any 
curves,  such  as  nb  and  aL",  parallel  to  a'U  represent 
isentropic  changes.  The  heat  rejected  along  ab  is  thus 
shown  by  abnn^  without  the  help  of  any  auxiliary 
diagram. 
a&U'S'r  represents  the  heat  received  per  cycle  in  the 


ACTUAL  STEAM-EXGIXE  CYCLE.  273 

W  pounds  of  cylinder  feed,  abcdef  represents  the  heat 
utilized,  so  that  the  difference,  M,  represents  the  total 
heat  losses.  These  consist  of  the  exhaust  -heat  ataee1} 
the  heat  lost  during  compression  abnn^,  and  the  heat 
lost  during  admission.  The  heat  refunded  during  ex- 
pansion is  represented  by  dee^.  Hence  the  heat  lost 
during  admission  equals 


M  —aiaee1  —abnn^  +dee^d  =  cL 


If  the  condition  of  the  steam  is  desired  at  any  point 
of  the  expansion  de,  it  is  found  by  reference  to  the  lines 
L  and  S  and  not  with  reference  to  L"  and  S'.  Hence 
the  initial  condensation  changed  the  state  point  from 
S  to  df,  so  that  this  loss  is  represented  by  the  area 
under  Sd'  and  not  by  that  under  S'd'.  Subtracting 
this  from  the  total  loss  during  admission  there  is  left 

cL"gcdd'  +&S1nS'  -njibgaa^ 

as  the  heat  losses  due  to  wire-drawing  and  friction. 

This  method  is  simple  and  easy  of  application,  as  it 
requires  the  construction  of  only  two  extra  curves,  L' 
and  S'.  Furthermore  it  gives  a  very  complete  account 
of  the  clearance  steam.  The  only  objection,  appar- 
ently, is  that,  with  the  exception  of  the  expansion  line 
de,  it  gives  an  entirely  false  idea  of  the  cycle  of  the  cyl- 
inder feed,  as  this  in  reality  is  entirely  condensed  and 
then  heated  along  aL"  in  the  boiler. 


274        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

If  the  cylinder  is  jacketed,  the  heat  given  out  by  the 
jacket  steam  may  be  indicated  at  the  right  of  XH 
(Fig.  97),  as  shown  under  XJ.  Further,  if  the  radiation 
loss  is  known,  it  may  be  represented  under  JR,  and 
then  the  remaining  area  under  XR  will  represent  the 
heat  given  by  the  jackets  to  the  steam  in  the  cylinder. 

Separate  Cycles  for  the  Cylinder  Feed  and  the  Clear- 
ance Steam. — Whatever  the  assumptions  made  with 
reference  to  the  card,  its  total  area  must  not  be  changed. 
Thus,  even  if  an  attempt  is  made  to  draw  separate 
cycles  for  both  cylinder  feed  and  clearance  steam,  the 
compression  line  of  the  resultant  cylinder-feed  card  will 
never  exactly  coincide  with  the  water  line  of  the  T<f>- 
diagram,  so  that  this  line  must  always  remain  imagi- 
nary hi  its  readings.  It  is,  however,  possible  to  have 
the  reconstructed  expansion  line  represent  the  true  pv- 
history  of  the  steam.  Thus,  in  place  of  drawing  an 
isentropic  line  through  the  point  of  compression,  draw 
the  polytropic  curve  pvn  =  C,  where  n  has  the  value 
found  for  the  expansion-curve.  Assuming  that  the 
clearance  steam  is  expanded  and  compressed  along  this 
line,  the  card  for  the  cylinder  feed  can  be  constructed 
by  assuming  this  curve  to  represent  zero  volume.  The 
7^-projection  of  the  expansion-curve  will  thus  repre- 
sent the  true  average  history  of  the  cylinder  feed,  and 
the  compression  line  will  follow  more  closely  the  water 
line  than  it  does  when  an  adiabatic  curve  is  taken  as 
the  new  base  line. 


ACTUAL  STEAM-ENGINE  CYCLE.  275 

The  cycle  for  the  clearance  steam  can  be  found  be- 
tween cut-off  and  release,  and  between  compression  and 
admission,  but  the  rest  of  it  would  be  entirely  imagi- 
nary. Possibly  it  might  be  continued  from  admission 
up  to  the  attainment  of  initial  pressure  by  assuming 
adiabatic  compression.  As  may  be  seen  from  Fig.  96, 
the  clearance  steam  contains  less  moisture  at  admission 
and  compression  than  at  cut-off  and  release  respec- 
tively, so  that  whatever  its  exact  path  between  admis- 
sion and  cut-off  and  between  release  and  compression, 
it  must  at  least  show  decreasing  and  increasing  values 
of  X  respectively,  so  that  its  history  is  the  reverse  of 
that  shown  by  the  indicator-card  itself. 

It  is  doubtful  if  the  increased  labor  involved  in  the 
making  of  such  a  plot  would  be  recompensed  by  any 
added  information  which  could  not  be  obtained  by  a 
proper  interpretation  of  the  simple  method  used  by 
Prof.  Boulvin. 


CHAPTER  XIV. 
STEAM-ENGINE    CYLINDER   EFFICIENCY. 

THE  thermal  efficiency  of  an  engine,  that  is,  the  frac- 
tional part  of  the  heat  energy  received  which  it  changes 
into  useful  work,  is  limited  first  by  the  ideal  efficiency 
which  is  fixed  as  soon  as  the  initial  quality  and  pres- 
sure and  the  back  pressure  are  known,  secondly,  by 
the  cylinder  efficiency  which  is  a  measure  of  the  heat 
losses  in  the  cylinder,  and  finally,  by  the  mechanical 
efficiency  which  is  a  measure  of  the  mechanical  fric- 
tional  losses  of  the  whole  engine  and  is  equal  to  the 
brake  horse-power  divided  by  the  indicated  horse- 
power. The  thermal  efficiency  is  thus  the  product 
of  these  three  separate  efficiencies,  or 

'Jtotal =  'jRankine  '  'Jcylinder  '  ^mechanical- 

In  order  to  make  the  total  thermal  efficiency  a 
maximum  the  engineer  must  so  design  the  engine 
that  each  of  the  component  efficiencies  possesses  its 
maximum  value.-  The  factors  influencing  the  Ran- 
kine  efficiency  have  already  been  discussed  in  Chapters 
XI  and  XII,  and  this  is,  in  any  case,  determined  by 

276 


STEAM-ENGINE  CYLINDER  EFFICIENCY.        277 

the  running  conditions  in  the  plant.  The  mechanical 
efficiency  is  controlled  by  workmanship  and  the  choice 
of  proper  metals  for  bearing  surfaces  and  of  suitable 
lubricants.  It  is  thus  a  mechanical  and  not  a  thermal 
problem,  and  so  falls  beyond  the  scope  of  this  treatise. 
The  cylinder  efficiency  being  the  result  of  purely 
thermal  processes  can,  as  shown  in  the  preceding 
chapter,  be  submitted  to  the  temperature-entropy 
analysis. 

In  order  to  minimize  the  heat  losses  in  a  steam- 
engine  it  becomes  necessary  to  understand  the  various 
factors  which  produce  them.  The  losses  are  of  three 
kinds : 

(1)  Loss   of   availability   due   to   throttling   during 
admission  and  exhaust; 

(2)  External  radiation  and  conduction  loss 

(3)  Condensation    and    re-evaporation    inside    the 
cylinder. 

The  throttling  losses  depend  upon  the  relative  areas 
of  piston  and  steam  ports,  as  well  as  upon  the  speed 
of  the  piston  and  the  length  and  crookedness  of  the 
ports.  In  the  case  of  compound  engines  the  length 
and  cross-sectional  areas  of  the  receivers  and  piping 
between  the  cylinders  must  also  be  considered.  For 
a  given  engine  the  higher  the  speed  the  greater  the 
throttling  loss. 

The  external  loss  depends  upon  the  difference  be- 
tween the  mean  temperature  of  the  cylinder  walls 


278       THE    TEMPERATURE-ENTROPY   DIAGRAM. 

and  the  outside  air,  upon  the  area  of  the  radiating 
surface  and  the  conductivity  of  the  materials  used. 
A  high-pressure  cylinder  will  radiate  more  heat  per 
unit  area  than  a  low-pressure  cylinder;  a  steam- 
jacketed  cylinder  will  lose  more  heat  externally  than 
a  non-jacketed  cylinder,  because  of  its  higher  tem- 
perature and  larger  surface.  This  loss  is  small  in 
comparison  with  the  condensation  loss  and  can  be 
minimized  by  lagging  the  cylinder  with  some  suitable 
insulating  material. 

The  condensation  and  re-evaporation  loss  is  caused 
by  the  tendency  of  heat  to  equalize  the  temperature 
of  bodies  in  thermal  contact.  Thus  the  cylinder  and 
piston  are  exposed  to  steam  fluctuating  in  tempera- 
ture between  that  corresponding  to  the  pressures  at 
admission  and  exhaust.  Investigations  have  shown 
that  the  fluctuations  in  temperature  of  the  metal 
extend  only  a  slight  distance,  two  or  three  hundredths 
of  an  inch,  into  the  mass  of  the  cylinder.  It  is  thus 
almost  a  surface  phenomenon  and  the  mass  of  metal 
participating  in  it  is  proportional  (practically)  to  the 
surfaces  exposed  to  the  steam.  The  quantity  of  heat 
transferred  is  proportional  to  the  difference  of  tem- 
perature between  the  steam  and  metal,  the  area,  and 
the  time.  Of  course  if  the  time  interval  is  long  enough 
the  metal  will  be  raised  from  the  temperature  of  ex- 
haust to  that  of  admission,  and  then  the  heat  required 
will  equal  the  weight  of  metal  involved  times  its  specific 


STEAM-ENGINE  CYLINDER  EFFICIENCY.        279 

heat  times  the  change  in  temperature,  but  at  higher 
speeds  the  temperature  of  the  steam  fluctuates  so 
rapidly  that  there  is  not  sufficient  time  for  the  metal 
to  arrive  at  either  the  temperature  of  admission  or 
exhaust,  and  it  thus  fluctuates  between  some  inter- 
mediate temperatures.  Another  way  of  looking  at 
this  is  to  assume  that  the  metal  directly  in  contact 
with  the  steam  assumes  its  temperature  immediately 
and  then  by  conduction  successive  layers  are  brought 
to  the  same  temperature.  This  requires  time,  so  that 
the  higher  the  speed  the  thinner  the  layer  of  metal 
which  is  involved  in  the  operation.  From  either  point 
of  view  the  only  way  to  eliminate  this  heat  transfer- 
ence would  be  to  run  the  engine  at  infinite  speed. 
Comparative  tests  made  upon  various  engines  by  setting 
the  governors  so  that  they  could  run  at  different  speeds 
with  the  same  conditions  of  cut-off,  and  of  initial  and 
back  pressures  always  show  that  the  condensation  loss  is 
nearly  proportional  to  the  time  required  for  a  revolution. 
Increased  speed  means,  however,  increased  port 
area  to  prevent  increased  throttling  loss  and  thus 
increased  area  of  radiating  surface  with  its  consequent 
loss.  The  number  of  revolutions  is  also  limited  by 
various  mechanical  factors,  so  that  the  speed  in  any 
given  case  may  be  considered  as  fixed.  It  therefore 
remains  to  investigate  the  effect  upon  initial  con- 
densation of  the  other  elements  of  design  at  the  dis- 
posal of  the  designer,  viz., 


280       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

(1)  Size  of  unit; 

(2)  Point  of  cut-off; 
-  (3)  Compounding; 

(4)  Steam  jacketing; 

(5)  Use  of  superheated  steam. 

(1)  The   mass    of   steam    contained   in   a   cylinder, 
other  things  being  the  same,  increases  as  the  cube  of 
the  dimensions,  while  the  area  exposed  to  the  steam 
Increases  only  as  the  square  of  the  dimensions.     Thus 
the  radiating  surface  per  unit  weight  of  steam  dimin- 
ishes as  the  first  power  with  increasing  size.     That  is, 
doubling   the   dimensions,    or  increasing   the    volume 
eight  times,  reduces  the  initial  condensation  per  pound 
of  steam  in  the  larger  cylinder  to  one-half  its  amount 
in  the  smaller  cylinder;    trebling  the   diameter  and 
stroke  reduces  the  condensation  to  one-third  its  origi- 
nal amount. 

(2)  The  position  of  cut-off  in  the  high-pressure  cylin- 
der controls   the  weight   of   steam   admitted   to   the 
cylinder  and  therefore  the  power  developed  per  stroke. 
The  longer  the  period  of  admission  the  greater  the 
horse-power.     Does  the  thermal  efficiency  or  economy 
increase  at  the  same  time,  or  does  it  fluctuate,  and 
if  so,  at  what  point  does  it  attain  its  maximum  value? 
Roughly  speaking,  the  cylinder  head  and  the  piston 
and  that  portion  of  the  cylinder  exposed  to  the  steam 
up  to  the  point  of  cut-off  have  their  temperatures 
raised  to  that  of  the  entering  steam,  while  the  tempera- 


STEAM-ENGINE    CYLINDER    EFFICIENCY.       281 

ture  of  the  rest  of  the  cylinder  is  raised  varying  amounts 
as  the  temperature  of  the  steam  drops  during  expan- 
sion. As  cut-off  changes  the  heat  required  to  warm 
the  metal  also  changes  slightly,  because  the  portion  of 
the  cylinder  heated  to  admission  temperature  changes 
with  the  cut-off,  but  a  large  part  of  the  surface,  viz., 
the  cylinder-head,  the  piston,  and  the  inside  of  the 
steam-ports,  remains  the  same,  so  that  although  the 
total  variation  in  cooling  surface  increases  and  de- 
creases with  the  cut-off  it  is  at  a  much  slower  rate. 
The  heat  required  to  warm  the  cylinder  per  stroke  is 
thus  but  slightly  variable,  being  nearly  the  same  for 
wide  variations  in  cut-off,  but  the  weight  of  steam 
increases  with  the  cut-off  so  that  the  loss  of  heat  per 
pound  diminishes  as  the  cut-off  increases.  The  maxi- 
mum condensation  loss  is  thus  found  at  early  cut-off 
and  the  minimum  loss  when  steam  is  taken  through 
out  the  entire  stroke.  Apparently  from  this  point  of 
view  the  economy  of  an  engine  increases  with  the 
power  developed. 

If  no  heat  losses  occurred  in  the  cylinder,  expansion 
from  early  cut-off  would  carry  the  pressure  at  the 
end  of  the  stroke  below  back  pressure,  then  as  cut-off 
increased  the  final  pressure  would  also  increase,  and  at 
some  definite  point  of  cut-off  would  just  equal  the 
back  pressure.  From  this  point  on  the  final  pressure 
would  exceed  the  back  pressure,  and,  finally,  when 
cut  off  occurred  at  the  end  of  the  stroke  there  would 


282       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


be  no  drop  of  pressure  whatever.  Due  to  the  con- 
densation during  the  first  part  of  the  expansion  the 
pressure  in  the  actual  cylinder  drops  more  rapidly 
than  during  adiabatic  expansion,  so  that  the  cut-off 
which  will  give  a  pressure  at  the  end  of  expansion  just 
equal  to  the  back  pressure  is  somewhat  later  in  the 
actual  than  in  the  theoretical  case.  If  cut-off  occurs 
later  than  this  the  pressure  at  release  drops,  without 
the  performance  of  work,  down  to  the  back  pressure; 
that  is,  the  availability  of  its  internal  energy  is  wasted. 


FIG.  99. 

Thus  the  more  cut-off  exceeds  that  value  at  which  the 
expansion  would  bring  the  pressure  down  to  exhaust 
pressure  the  greater  the  wasted  internal  energy.'  Con- 
sidered from  this  aspect  alone  the  engine  would  first 
run  with  a  negative  loop  in  the  card,  so  that  its  economy 
would  increase  up  to  the  point  at  which  the  loop  dis- 


STEAM-ENGINE    CYLINDER    EFFICIENCY.      283 

appeared,  and  from  this  point  onwards  the  economy 
would  decrease  slowly  as  the  power  increased. 

There  are  thus  two  factors  exerting  opposing  in- 
fluences: initial  condensation  which  diminishes,  and 
loss  of  internal  energy  which  increases  with  the  load. 
The  mutual  variations  of  these  factors  cannot  in  the 
present  state  of  our  knowledge  be  predicted  from 
point  to  point,  but  it  is  at  least  evident  that  there  is 
some  cut-off  at  which  the  sum  of  these  two  losses  will 
prove  a  minimum  (Fig.  99).  To  determine  this  point 
of  most  economical  cut-off  for  any  given  engine  economy 
tests  must  be  run  at  varying  loads  for  the  same  boiler 
and  back  pressures.  Then  the  efficiency  and  cut-off 
must  be  plotted  as  coordinates  and  the  minimum  point 
of  the  curve  thus  established  will  be  the  desired  cut-off. 

(3)  The  heat  absorbed  by  the  cylinder  metal  each 
revolution  is  proportional  to  the  change  of  tempera- 
ture which  the  metal  undergoes;  this  in  turn  depends 
upon  the  difference  in  temperature  of  the  entering  and 
leaving  steam.  Therefore  the  greater  the  range  of 
pressure,  or  better,  of  temperature,  in  a  cylinder  the 
greater  the  initial  condensation.  Halving  the  tempera- 
ture range  halves  the  condensation  loss.  Thus  if  an 
engine  is  to  work  between  fixed  temperature  limits 
the  initial  condensation  loss  may  be  reduced  by  sub- 
dividing this  temperature  drop  among  several  cylinders, 
that  is,  by  compounding  the  engine.  The  steam 
condensed  in  the  first  cylinder  is  re-evaporated  during 


284      THE  TEMPERATURE-ENTROPY  DIAGRAM. 


exhaust  and  it  can  thus  be  utilized  in  the  second 
cylinder  to  produce  power  or  to  heat  the  walls.  The 
same  steam  could  be  used  over  and  over  in  successive 
cylinders  to  heat  the  walls,  so  that  the  condensation 
losses  are  not  additive.  Therefore  the  greater  the 
number  of  cylinders  the  smaller  the  condensation  loss. 
But  the  addition  of  each  new  cylinder  entails  an 
extra  throttling  loss  in  its  admission-  and  exhaust-ports, 
an  extra  external  radiation  loss  from  its  walls  and  from 
the  piping  and  receiver  which  connect  it  to  the  pre- 
ceding cylinder.  These  radiation  and  transference 
losses  are  practically  proportional  to  the  number  of 
cylinders. 

/  \ 


L   =£ 


FIG.  100. 


In  compounding  there  are  thus  two  opposing  factors 
to  be  considered:  the  initial  condensation  loss  which 
diminishes,  and  the  transference  and  radiation  losses 
which  increase  with  the  number  of  cylinders.  As 


STEAM-ENGINE  CYLINDER    EFFICIENCY.       285 

before,  the  theoretical  development  of  the  subject 
proves  inadequate  and  the  proper  degree  of  compound- 
ing for  different  pressure  ranges  must  be  determined 
experimentally. 

Fig.  100  sh:nvs  the  character  of  these  losses  for  simple, 
compound,  and  triple  expansion,  between  the  same 
initial  and  final  conditions. 

(4)  Watt  stated  that  the  function  of  a  steam-jacket 
was  to  heat  the  engine  cylinder  to  the  temperature 
of  the  entering  steam.  If  a  jacket  entirely  fulfilled 
this  requirement  initial  condensation  of  steam  inside 
a  cylinder  would  be  entirely  eliminated.  This  would 
mean  that  the  cooling  of  the  cylinder  walls  by  low- 
temperature  steam  during  expansion  and  exhaust  must 
be  made  up  before  admission  by  heat  taken  from  the 
jacket.  During  expansion  the  steam  would  receive 
heat  from  the  hotter  walls  and  its  pressure  would  thus 
be  maintained  at  a  higher  value  than  that  correspond- 
ing to  adiabatic  expansion.  Some  work  would  thus 
be  performed  by  the  heat  received  from  the  jacket. 
But  this  heat,  thus  received  at  low  temperatures,  could 
not  be  as  efficient  as  the  high-temperature  heat  contained 
i:i  the  entering  steam.  During  exhaust  the  walls, 
being  at  the  temperature  of  live  steam,  give  heat  directly 
to  the  steam  which  heat  is  carried  away  without  per- 
forming any  work.  In  fact  by  increasing  the  volume 
of  the  exhaust  steam  it  may  actually  increase  the 
throttling  loss  during  exhaust.  The  steam-jacket  by 


286       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

increasing  the  external  temperatures  and  dimensions 
of  the  cylinder  also  increases  the  external  radiation 
and  conduction  losses. 

The  steam-jacket  is  thus  not  only  an  inefficient 
method  for  supplying  heat  to  the  inside  of  the  cylinder 
for  the  development  of  work,  but  it  also  increases  the 
external  losses.  The  increased  economy  obtained  by 
the  use  of  a  jacket  indicates  that  the  decrease  in 
initial  condensation  must  more  than  offset  the  losses 
inherent  in  the  jacket.  The  reason  for  this  is  readily 
found.  Although  steam  is  ordinarily  nearly  dry  at 
the  throttle  the  moisture  at  cut-off  may  range  any 
where  from  25  per  cent,  to  50  per  cent.  The  steam  at 
release  usually  contains  somewhat  less  moisture  than 
at  cut-off,  but  not  to  any  great  amount.  The  re- 
evaporation  of  this  moisture  during  exhaust  extracts  a 
large  quantity  of  heat  from  the  metal  which  must  be 
be  made  up  by  the  entering  steam.  If  now  the  jacket 
by  preventing  initial  condensation  permits  of  dry 
steam  at  cut-off,  a  lesser  weight  of  steam  would  be  re- 
quired and  at  release  this  would  not  contain  more 
moisture  per  pound  than  that  naturally  incident  to 
adiabatic  expansion.  There  would  thus  be  a  smaller 
quantity  of  water  to  be  evaporated  during  exhaust, 
and  consequently  correspondingly  less  heat  would  be 
required  from  the  cylinder  walls.  Further,  all  the  heat 
.lost  to  the  cylinder  walls  through  initial  condensation 
and  re-evaporation  is  entirely  wasted,  and  only  the 


STEAM-ENGINE   CYLINDER    EFFICIENCY.      287 

heat  of  the  liquid  at  exhaust  temperature  can  be  returned 
to  the  boiler,  but  some  of  the  heat  given  to  the  cylinder 
by  the  jacket  does  work  during  expansion  and~  the 
heat  of  the  liquid  can  be  restored  to  the  boiler  at  boiler 
temperature.  Experience  has  shown  that  in  most  cases 


CB 


FIG.  101. 

a  net  saving  is  obtained  from  the  use  of  a  jacket.  Natu- 
rally, the  engine  having  the  largest  condensation  loss 
would  be  most  benefited  by  the  addition  of  a  jacket. 
Thus  a  jacket  would  produce  a  greater  saving  on  a 
small  engine  than  on  a  large  one,  on  a  slow-speed 
engine  than  on  a  high-speed  (speed  refers  to  number 
of  revolutions  and  not  the  distance  traversed  by  the 


288      THE  TEMPERATURE-ENTROPY  DIAGRAM. 

piston  in  feet  per  minute),  at  early  cut-off  than  at 
late  cut-off. 

There  is  of  course  no  expression  for  ideal  efficiency 
involving  jacket  steam,  as  heat  conduction  losses  do 
not  exist  in  an  ideal  engine,  but  in  any  actual  engine 
the  thermal  efficiency  of  the  indicated  horse-power  is 
given  by 

__  2545XI.H.P. 

I?I.H.P.->?R.  ^cylinder  -gteam  per  hour-  (#1  -g2)  +  jacket 

steam  per  hourX?"i 
2545 


where  wc  and  w}-  are  cylinder  feed  and  jacket  steam  per 
I.H.P.  per  hour  respectively. 

If  the  card  is  projected  into  the  T^-plane  every- 
thing is  established  upon  the  basis  of  1  Ib.  of  cylinder 
feed,  so  that  the  expression  for  the  efficiency  would  be 

Area  of  card  in  T^-plane 


It  is  thus  possible  to  indicate  the  heat  received  from 
the  jacket  steam  by  laying  off  a  rectangle  ab,  Fig.  101, 

equal  to  —-r\.     If  the  external  radiation  loss  is  known 

Wc 

this  can  be  further  subdivided  into  ac  and  cb,  repre- 
senting heat  given  to  the  cylinder  feed  and  that  lost 
in  radiation  respectively. 


STEAM-ENGINE   CYLINDER   EFFICIENCY.       289 

The  cylinder  efficiency  can  be  obtained  from  the 
above  by  dividing  by  the  Rankine  efficiency. 

When  jackets  are  used  on  intermediate  and  low- 
pressure  cylinders  the  jacket  steam  is  at  a  higher 
temperature  than  the  cylinder  feed  during  admission 
and  can  thus  more  readily  perform  its  function. 

The  use  of  jackets  on  and  reheating  coils  in  inter- 
mediate receivers  has  the  primary  object  of  drying 
the  exhaust  from  the  preceding  cylinder  before  it  passes 
on  to  the  next.  If  the  steam  is  not  very  wet,  or  if  the 
moisture  is  removed  by  a  separator,  they  may  serve 
to  superheat  the  steam.  Certain  tests  seem  to  indicate 
that  unless  they  superheat  the  steam  they  fail  in  their 
purpose.* 

(4)  As  mentioned  on  pp.  236-239  the  use  of  super- 
heated steam  has  but  slight  effect  upon  the  Rankine 
efficiency.  Thus  with  steam  pressure  of  200  Ibs. 
absolute  and  exhaust  at  1  Ib.  absolute  the  Rankine 
efficiency  for  dry  saturated  steam  is  30.0  per  cent., 
while  with  steam  superheated  300°  F.  the  efficiency 
is  only  31.3  per  cent.  The  large  saving  actually 
effected  by  superheated  steam  must  therefore  be  due 
to  its  effect  upon  the  cylinder  efficiency.  The  explana- 
tion is  found  in  the  observed  fact  that  heat  is  conducted 
less  rapidly  between  a  gas  and  a  metal  than  between 
a  liquid  and  a  metal.  Thus  Ripper  (Steam,  Engine 
Theory  and  Practice,  p.  149)  found  in  a  certain  small 

*  Trans.  A.  S.  M.  E.,  vol.  xxv,  pp.  443-501. 


290       THE  TEMPERATURE-ENTROPY  DIAGRAM 


engine  "that  for  each  1  per  cent,  of  wetness  at  cut-off, 
7.5°  F.  of  superheat  must  be  present  in  the  steam  on 
admission  to  the  engine  to  render  the  steam  dry  at 


FIG.  102. 

cut-off."  The  loss  of  a  small  amount  of  superheat 
prevents  a  larger  condensation  due  to  the  decreased 
rate  of  flow.  And  furthermore  the  steam  containing 
less  moisture  at  release,  less  heat  is  removed  from  the 


STEAM-ENGINE   CYLINDER   EFFICIENCY.       291 

cylinder  metal,  and  consequently  less  heat  is  required 
from  the  next  admission  steam. 

Let  1,  Fig.  102,  represent  the  expansion  line  when 
dry-saturated  steam  is  supplied  to  the  engine  and  2 
when  sufficient  superheat  is  present  to  give  dry  steam 
at  cut-off.  The  cross-hatched  portion  represents  the 
extra  heat  utilized  per  pound  because  of  the  addition 
of  the  small  amount  of  superheat.  This  represents 
a  case  where  the  heat  received  per  pound,  H1—q2,  is 
increased  by  about  one-fifth  or  one-sixth,  while  the 
heat  utilized  per  pound  is  nearly  doubled. 

Comparison  between  Reciprocating  Engines  and  Tur- 
bines.— In  the  steam  turbine  there  is  nothing  analogous 
to  the  initial  condensation  and  re-evaporation  losses 
of  the  reciprocating  engine.  After  the  turbine  is 
once  in  operation  a  constant  temperature  gradient  is 
established  from  the  admission  to  the  exhaust.  That 
is,  the  steam  is  continually  diminishing  in  temperature 
as  its  pressure  drops  in  the  successive  nozzles  and  stages, 
and  the  adjacent  metal  soon  acquires  the  same  tempera- 
ture as  the  steam.  Thus  the  heat  conduction  losses 
are  of  two  kinds:  transference  of  heat  through  the 
metal  from  the  steam  to  the  atmosphere  and  trans- 
ference of  heat  through  the  metal  by  conduction  from 
steam  of  high  temperature  to  steam  of  lower  tempera- 
ture. Both  of  these  losses  are  probably  relatively  small 
as  compared  with  the  condensation  and  re-evaporation 
losses  of  the  reciprocating  type. 


292        THE  TEMPERATURE-ENTROPY   DIAGRAM. 


Expansion  in  the  reciprocating  engine  must  not  be 
carried  beyon  I  the  point  at  which  the  difference  of 
pressure  upon  the  two  faces  of  the  piston  becomes 
less  than  the  frictional  resistance  of  the  engine,  as 
such  an  operation  would  mean  the  wasting  of  power 
already  realized.  In  other  words,  part  of  the  heat 
theoretically  available  for  work  must  be  sacrificed  to 
avoid  a  greater  friction  loss.  In  the  turbine,  however, 


FIG.  103. 

no  such  limitation  is  placed  upon  the  expansion.  In 
fact  the  greater  the  expansion  in  a  turbine  the  smaller 
the  friction  loss,  as  the. major  part  of  the  friction  is 
clue  to  windage,  and  this  decreases  as  the  density  of 
the  steam  surrounding  the  rotor  diminishes.  It  would 
thus  seem  t'hat  the  turbine  has  a  distinct  advantage 
over  the  reciprocating  type,  as  complete  advantage 
may  be  taken  of  the  theoretical  possibilities  by  expand- 
ing completely  down  to  back  pressure  with  a  net  saving 
of  the  heat  area  abc,  Fig.  103. 


STEAM-ENGINE  CYLINDER  EFFICIENCY.          293 

In  the  reciprocating  engine  the  throttling  losses  during 
admission  and  exhaust  are  comparatively  small,  and 
exist  primarily  because  of  the  impossibility  of  making 
the  ports  large  enough  consistent  with  good  design 
to  prevent  drop  in  pressure.  In  the  turbine  type,  on 
the  other  hand,  wire  drawing  occurs  because  of  the 
impossibility  of  making  the  clearance  spaces  about 
the  blades  small  enough  to  prevent  it.  Thus  in  each 
stage  a  considerable  portion  of  the  steam  passes  by 
the  blades  without  performing  work  upon  them. 

Each  type  of  motor  thus  possesses  its  own  method 
of  dissipating  the  availability  of  the  heat  in  the  entering 
steam,  and,  judging  from  economy  tests,  the  total  losses 
are  about  the  same  in  both  cases. 


CHAPTER  XV. 
LIQUEFACTION   OF   VAPORS   AND   GASES. 

SUPERHEATED  steam  of  the  condition  shown  at  a  in 
Fig.  104  might  be  changed  to  saturated  steam  by  one 


FIG.  104. 

of  two  methods.  First,  it  could  be  kept  in  a  hot  bath 
at  constant  temperature  and  the  pressure  increased 
from  pa  to  7)5,  so  that  its  state  point  would  move  from 
a  to  6.  Secondly,  it  might  be  kept  under  constant  pres- 
sure pa  (as  a  weighted  piston)  and  its  temperature 
allowed  to  drop  by  radiation  from  ta  to  tc)  so  that  the 
state  point  travels  from  a  to  c.  Liquefaction  mil 


LIQUEFACTION  OF  VAPORS  AXD  GASES.         295 

begin  by  either  process  as  soon  as  the  state  point  reaches 
the  dry-steam  line. 

It  is  at  once  evident  that  the  second  method  is  the 
only  one  always  applicable,  for  the  isothermal  change 
might  take  place  above  the  critical  temperature  and 
then  no  increase  of  pressure,  however  great,  could 
result  in  liquefaction. 

The  critical  temperature  of  steam  is  beyond  the 
upper  limit  of  temperature  used  hi  engineering,  so 
that  this  is  not  as  clear  here  as  in  the  case  of  some 
vapor  which  superheats  at  ordinary  temperatures,  as, 
for  example,  carbon  dioxide. 

Fig.  105*  shows  the  T$-,  pv-,  Tp-,  and  v^-diagrams  for 
C02.  The  chief  differences  between  this  diagram  and 
that  for  steam  lie  in  the  fact  that  the  critical  tempera- 
ture is  included,  thus  showing  the  intersection  of  the 
liquid-  and  saturated-vapor  curves,  and  further,  that 
the  volume  of  the  liquid  is  now  appreciable  with  refer- 
ence to  that  of  the  vapor  and  its  variation  with  increas- 
ing pressure  and  temperature  no  longer  negligible. 

If  a  (Fig.  105)  represent  the  state  point  of  the  super- 
heated C02  vapor  at  the  pressure  and  temperature 
under  consideration,  isothermal  compression  will  fail  to 
produce  condensation,  although  cooling  at  constant  pres- 
sure will  produce  liquefication  as  soon  as  c  is  reached. 

*  This  diagram  is  only  approximately  correct,  being  based 
upon  somewhat  discrapant  data  given  by  Amagat,  Regnault, 
and  Zcuner. 


296        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

The  ordinary  gases,  hydrogen,  oxygen,  nitrogen,  etc., 
have    their   critical   points   as   much   below    ordinary 


FIG.  105. 

engineering  temperatures  as  that  of  water  is  above 
them. 


LIQUEFACTION  OF  VAPORS  AXD  GASES.         297 

Superheated  steam  at  ordinary  temperatures  could 
be  liquefied  isothermally  because  its  temperature  is  less 
than  that  of  the  critical  point;  carbonic  dioxide  at 
ordinary  atmospheric  temperatures  could  also  ordinarily 
be  liquefied  isothermally  because  the  usual  atmospheric 
temperature  is  less  than  31. 9°  C.  (89.4°  F.);  hydrogen, 
oxygen,  etc.,  cannot  be  liquefied  isothermally  simply 
because  the  atmospheric  temperature  is  far  above 
their  critical  temperatures.  The  essential  factor  in 
liquefaction,  then,  is  to  reduce  the  temperature  and  then 
simply  to  compress  the  gas  isothermally  until  lique- 
faction commences. 

Suppose  it  is  desired  to  liquefy  some  carbon  dioxide 
some  summer  day  when  the  temperature  of  the  atmos- 
phere is  above  its  critical  temperature.  Let  the  com- 
pression be  carried  on  slowly,  so  that  the  heat  generated 
may  be  dissipated  by  radiation  and  the  process  be 
isothermal.  Liquefaction  will  not  occur.  It  will  be 
necessary  to  cool  the  gas  down  to  the  critical  tempera- 
ture by  some  means,  physical  or  chemical.  Possibly 
a  coil  containing  cold  water  will  suffice  in  this  case. 
Proceeding  to  other  substances  possessing  lower  and 
lower  critical  temperatures,  cooling  mixtures  giving 
lower  and  lower  temperature  would  be  required.  That 
is,  by  this  method  it  would  never  be  possible  to  liquefy 
any  substance,  however  great  the  pressure  applied, 
unless  there  already  existed  some  source  of  cold  as 
low  as  its  critical  temperature.  When  the  "permanent " 


298        THE  TEMPERATURE-EXTROPY  DIAGRAM. 


gases  are  reached  the  sources  of  artificial  cold  fail,  and 
unless  the  gas  may  be  made  to  cool  itself  investigation 
must  cease.  Any  new  gas  thus  liquefied  of  course  in 
turn  becomes  a  new  source  of  cold  to  aid  in  further 
investigation. 

Let  Fig.  106  represent  the  T^-diagram  of  some  gas 
having    a    low    critical    temperature.    Consider    the 


FIG.  106. 


throttling-curve  abc.  By  definition  this  represents  an 
adiabatic  change,  during  which  no  work  is  performed; 
i.e.,  the  heat  contained  in  the  substance  is  a  constant. 
For  this  curve  the  first  law  of  thermodynamics  gives 


which  becomes 


LIQUEFACTION  OF  VAPORS  AND  GASES.         299 

that  is,  the  curve  represents  an  irreversible  isodynamic 
process  in  the  case  of  a  perfect  gas  and  differs  but 
slightly  from  it  for  ordinary  gases. 

Now,  the  internal  energy  of  any  substance  is  defined 
as  the  summation  of  the  sensible  heat  and  the  dis- 
gregation  work,  or  the  kinetic  energy  of  the  molecules 
due  to  their  own  vibrations  plus  the  potential  energy 
due  to  their  mutual  positions.  Representing  these 
quantities  by  S  and  /  respectively,  it  follows  that 
Si  +A  -S2  -L  =  p2v2  -piVi. 

The  work  necessary  to  separate  the  molecules  against 
their  mutual  attraction  must  increase  with  the  distance 
between  them  although  the  rate  of  increase  is  inversely 
as  the  square  of  the  distance  between  them. 

As  the  volume  increases  the  value  of  /  increases 
and  hence  the  value  of  8  must  decrease  an  amount 
equal  to 


that  is,  the  temperature  of  the  substance  decreases.* 
With  increasing  volume  the  rate  of  temperature  drop 
decreases  so  that  the  curve  abc  approaches  T=  const. 
as  an  asymptote.  Near  the  saturation  curve,  in  the 
region  of  the  "superheated"  vapors,  this  drop  is  con- 
siderable, but  far  to  the  right  of  this  curve  and  above 
the  critical  temperatures  these  "  throttling"  curves 
become  almost  parallel  to  the  ^-axis.  The  "perfect 

*For  hydrogen  at  ordinary  temperatures  ;y2-p,f,  is  negative 
and  greater  than  7a—  Ilt  so  that  the  temperature  increases. 


300        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

gas"   is  simply  the  limiting  condition  in   which  the 
potential  energy  has  attained  its  maximum  value,  or 
rather  where  any  limited  change  in  volume  does  not 
affect  the  total  value  of  /  appreciably. 
In  such  a  case 


and 


that  is,  the  disgregation  change  being  negligible,  the 
isodynamic  curves  become  coincident  with  the  iso- 
thermals. 

In  Fig.  106  let  at,  a2,  a3,  etc.,  represent  a  series  of  such 
throttling-curves.  Let  a  pound  of  air  be  taken  from 
its  initial  condition  P  (representing  atmospheric  pres- 
sure and  temperatifre)  and  be  compressed  isothermally 
to  1.  This  may  be  effected  by  jacketing  the  cylinder 
walls  of  the  compressor  with  cold  water.  If  the  air 
is  then  permitted  to  expand  along  the  throttling  -curve 
a1}  the  temperature  will  drop,  say,  from  1  to  2.  The 
air  at  reduced  pressure  is  fed  back  to  some  interme- 
diate stage  of  the  compressor.  In  some  forms  of  lique- 
fiers  the  expansion  is  carried  at  once  down  to  atmos- 
pheric pressure,  thus  securing  a  somewhat  greater  drop 
in  temperature,  and  the  air  is  then  returned  to  the 
first  stage  of  the  compressor.  If  this  cooled,  expanded 
air  be  made  to  flow  back  outside  the  pipe  containing 
more  air  from  the  compressor,  this  in  turn  will  be 


LIQUEFACTION  OF  VAPORS  AND  GASES.         301 

cooled  by  conduction  to  some  lower  temperature  and 
so  will  escape  from  the  throttling- valve  at  pressure  pt 
at  some  lower  temperature  (3).  This  will  now  expand 
along  the  throttling-curve  a2  to  a  still  lower  tempera- 
ture (4).  This  process  is  continued  until  the  tempera- 
ture of  the  issuing  jet  has  fallen  below  the  critical 
temperature,  when  liquefaction  will  ensue. 

This  will  evidently  occur  first  in  the  nozzle,  as  the 
temperature  outside  will  need  to  be  still  further  de- 
creased before  the  air  will  remain  liquid  at  the  reduced 
pressure.  The  vaporization  of  the  liquid  first  formed 
tends  to  decrease  still  further  the  temperature  of  the 
air-tubes,  etc.  It  is  probable  that  at  first  all  of  this 
liquid  vaporizes  as  soon  as  ejected,  but  the  vaporiza- 
tion of  part  soon  cools  down  the  rest  and  its  surround- 
ings, so  that  a  small  portion  remains  liquid  at  the 
lower  pressure.  The  back  pressure  may  thus  in  time 
be  reduced  to  that  at  P  and  then  the  temperature  will 
be  found  at  which  air  vaporizes  at  atmospheric  pres- 
sure. 

The  expansion  of  the  air  thus  provides  in  itself  the 
cooling  process  needed  to  reduce  the  temperature 
below  the  critical  point  so  that  sufficient  increase  of 
pressure  may  cause  liquefaction. 


CHAPTER  XVI. 

APPLICATION    OF    THE    TEMPERATURE-ENTROPY  DIA- 
GRAM TO  AIR-COMPRESSORS  AND  AIR-MOTORS. 

The  Reversed,  Power-absorbing,  Thermodynamic  Pro- 
cesses.—  In  the  direct  power  producing  thermody- 
namic  processes  already  considered,  whether  the  simple 
theoretically  perfect  Carnot  cycle,  or  the  various  prac- 
tical cycles  such  as  the  Rankine  cycle  for  saturated 
and  superheated  vapors,  or  the  Ericsson  and  Stirling 
cycles  for  air,  or  the  Otto,  Joule  or  Diesel  cycles  for 
internal- combustion  engines,  the  fundamental  principle 
of  operation  is  the  same;  namely,  during  the  expan- 
sion of  the  working  fluid  heat  is  absorbed  at  highest 
possible  temperatures,  and  during  the  compressive 
stroke  heat  is  rejected  at  the  lowest  possible  tempera- 
tures, the  difference  between  the  amount  of  heat 
absorbed  and  the  heat  rejected  being  transformed  into 
mechanical  energy. 

In  all  of  these  processes  the  amount  of  work  produced 
by  means  of  the  cycle  is  always  much  less  than  the 
energy  supplied  to  the  working  substance,  because  a 
necessary  condition  for  the  transference  of  heat  into 
mechanical  work  is  a  temperature  difference.  The 
greater  this  temperature  difference  the  greater  the 

302 


AIR-COMPRESSORS  AND  AIR-MOTORS.  303 

fractional  part  of  the  heat  which  can  be  transferred 
into  work.  This  temperature  difference  is  therefore 
limited  by  the  range  in  temperature  between  that  at 
which  the  heat  is  supplied  and  that  of  the  coldest  of 
the  surrounding  bodies,  which  ordinarily  is  that  of  the 
atmosphere  or  of  the  cooling  water. 

In  all  these  cycles  the  maximum  efficiency  seldom 
ranges  above  30  per  cent,  more  ordinarily  25  per  cent, 
but  under  exceptional  conditions  the  Carnot  efficiency 
may  reach  50  per  cent.  In  the  actual  engine  working 
approximately  on  any  of  these  cycles  the  fractional  part 
of  the  heat  received  which  is  changed  into  work  is 
naturally  considerably  less  than  the  fractional  part 
theoretically  available,  due  to  heat  losses  of  varying 
kinds  and  magnitude.  In  other  words,  we  may  sum 
up  the  properties  of  these  cycles  as  follows: 

Heat  is  taken  in  at  high  temperature;  a  portion  of 
this  heat  is  changed  into  work,  but  most  of  it  is  rejected 
as  heat  at  lower  temperatures. 

In  the  reversed  power  absorbing  thermodynamic 
cycles  the  working  fluid  is  made  to  traverse  its  ther- 
modynamic history  in  a  reverse  direction;  namely, 
it  takes  in  heat  at  low  temperatures  while  expanding, 
is  then  compressed  and  gives  out  heat  at  high  tempera- 
tures, the  power  required  to  perform  this  compression 
being  equal  to  the  power  which  in  the  direct  cycle  is 
produced  by  the  expansion  of  the  working  fluid. 

In  the  usual  definition  for  efficiency  the  efficiency 


304       THE  T  EM  PER  AT  V  RE-ENTROPY  DIAGRAM. 

is  stated  as  output  divided  by  input.  For  the  direct 
cycles  this  magnitude  is  always  less  than  unity,  but  for 
the  indirect  cycles  its  value,  theoretically  at  least,  is 
always  greater  than  unity,  although  in  the  actual 
working  mechanism  heat  and  friction  losses  may 
serve  to  reduce  it  to  practically  unity,  and  sometimes 
even  less. 

As  we  analyze  this  reversed  cycle  we  find  that  the 
output  may  be  of  two  different  kinds,  i.e.,  the  cycle 
may  be  utilized  to  remove  heat  at  low  temperatures 
from  some  confined  space  (as  for  example  the  ordinary 
refrigerating  machinery),  or  may  be  utilized  to  deliver 
heat  at  high  temperatures  to  some  confined  space,  as 
in  the  ordinary  heating  and  ventilating  systems.  In 
both  of  these  cases  the  input  represents  the  work 
required  to  drive  the  compressor,  the  output,  however, 
corresponds  in  the  first  case  to  the  exhaust  heat  of  the 
direct  cycle,  and  in  the  second  case  to  the  ht  at  received 
in  the  direct  cycle.  Thus,  for  example,  in  the  use  of 
steam,  the  efficiency  of  the  Rankine  cycle  under  ordinary 
conditions  is  approximately  20  per  cent.  Therefore 
the  exhaust  heat  represents  80  per  cent  of  the  total 
heat  available  for  supply  to  the  engine.  In  other 
words,  the  exhaust  heat  is  four  times  the  magnitude 
of  the  utilized  heat.  Therefore  in  the  reversed  cycle, 
if  we  consider  the  cooling  effect  as  useful  output,  it  is 
evident  that  this  cooling  effect  is  four  times  the  mag- 
nitude of  the  force  required  to  drive  the  compressor, 


AIR-COMPRESSORS  AND  AIR-MOTORS.  305 

while  if  we  consider  the  heating  effect  of  the  exhaust 
steam  at  high  temperature,  it  would  have  five  times 
the  magnitude  of  the  power  required  to  drive  the  com- 
pressor. Again  the  suction  of  the  compressor  might 
be  used  for  refrigeration  and  its  discharge  for  heating, 
and  in  such  a  case  the  effective  thermal  output  or 
useful  effort  would  be  nine  times  that  required  to 
drive  the  compressor. 

A  third  use  of  the  compressor  is  the  production  of 
gas  under  high  tension  for  future  use  in  power  produc- 
tion, physical  experimentation,  and  industrial  use. 

The  Compressor. — In  all  of  these  systems  the  com- 
pressor plays  an  essential  part,  and  it  is  therefore  well 
to  make  a  study  of  the  compressor  cycle  first,  before 
entering  into  the  details  of  the  different  cycles  embody- 
ing this  -reversed  process.  The  compressor  is  really 
an  engine  driven  backward,  with  all  of  its  parts,  valve 
gears,  etc.,  operating  in  a  reverse  order.  The  usual 
cycle  of  operation  is  something  as  follows: 

Consider  the  piston  to  be  at  the  beginning  of  the 
stroke;  a  certain  small  quantity  of  air  is  compressed 
in  the  clearance  space  under  the  pressure  prevailing  in 
the  delivery  pipe.  As  the  piston  moves  forward  this 
clearance  air  expands  behind  it  until  the  pressure  has 
decreased  slightly  below  that  in  the  suction  pipe. 
The  small  difference  of  pressure  then  prevailing  between 
the  suction  pipe  and  the  inside  of  the  cylinder  suffices 
to  lift  the  suction  valve  off  its  seat  against  the  spring 


306       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

which  usually  holds  it.  As  the  piston  continues  to 
move  forward  new  air  will  rush  in  behind  it  to  the  end 
of  the  stroke.  It  is  quite  possible  that  the  inertia 
of  the  air  in  the  suction  pipe  will  be  such  that  even  after 
the  piston  slows  down  at  the  end  of  the  stroke  the  air 
will  continue  to  rush  into  the  cylinder,  and  may  even 
bring  the  pressure  in  the  cylinder  up  to  that  prevailing 
in  the  suction  pipe.  The  admission  valve  now  closes 
under  the  action  of  the  spring.  As  the  piston  starts 


FIG.  107. 

to  retrace  its  motion  it  crowds  the  air  ahead  of  it  into 
the  further  end  of  the  cylinder,  so  that  the  pressure 
continually  rises  until  it  reaches  such  a  magnitude 
as  to  slightly  exceed  that  in  the  delivery  pipe;  when 
the  small  difference  of  pressure  thus  created  suffices 
to  life  the  discharge  valve  from  its  seat,  and  then  during 
the  rest  of  the  stroke  the  air  will  be  delivered  into  the 
discharge  pipe.  This  cycle  repeats  itself  indefinitely. 
Referring  to  Fig.  107,  the  cycle  is  as  follows :  From  a  to 
d  the  gas  (or  vapor)  in  the  clearance  space  expands 
until  the  pressure  at  d  falls  sufficiently  below  that  in 
the  supply  pipe  to  permit  the  admission  valve  to  open 


AIR-COMPRESSORS  AND   AIR-MOTORS.  307 

and  admit  a  new  supply  from  d  to  c.  On  the  return 
stroke,  the  entire  quantity  is  compressed  along  cb, 
until  the  pressure  becomes  sufficient  to  lift  the  release 
valve,  when  discharge  occurs  from  &  to  a  against  the 
upper  pressure.  The  indicator-card  thus  shows  the 
entire  cycle  of  the  clearance  gas,  but  only  one  portion, 
the  compression,  of  that  of  the  charge.  The  expansion 
and  compression  of  a  gas  in  a  cylinder  is  more  nearly 
adiabatic  than  that  of  a  saturated  vapor,  and  hence, 
as  the  temperature  of  the  gas  at  the  end  of  admission 
is  nearly  that  existing  at  the  end  of  expansion,  the 
expansion  and  compression  of  the  gas  in  the  clearance 
space  may  be  considered  to  neutralize  each  other 
thermodynamically;  although  mechanically  the  greater 
the  clearance  the  greater  the  size  of  the  cylinder  neces- 
sary to  compress  a  given  amount  of  gas.  Therefore 
all  discussion  of  the  effects  of  clearance  will  be  omitted 
for  the  present. 

During  the  expansion  of  the  clearance  air  and  the 
compression  of  the  total  charge,  the  heat  interchanges 
between  air  and  cylinder  walls  are  much  smaller  than 
the  corresponding  changes  between  the  steam  and 
cylinder  walls  in  steam  engines,  i.e.,  the  operations 
are  nearly  adiabatic.  If  atmospheric  air  were  simply 
a  mixture  of  oxygen  and  nitrogen  the  equation  of  such 
a  compression  or  expansion  would  be  pvl'405  =  C,  but 
due  to  the  fact  that  a  certain  amount  of  moisture 
is  always  present  in  the  air  the  value  of  the  exponent 


308     THE   TEMPERATURE-ENTROPY  DIAGRAM. 

is  somewhat  different.    Thus  diatomic  gases,  as 
#2,  #2,  etc.,  possess  a  ratio  of  specific  heats 


—  =  1.4,  approximately, 


but  as  the  complexity  of  the  molecular  composition 
increases  the  value  of  this  ratio  decreases.  Thus  for 
superheated  H^O  the  value  is  approximately  1.31  to 
1.33,  so  that  the  mixture  of  atmospheric  air  and  super- 
heated water  vapor  gives  a  value  of  this  ratio  which 
is  something  less  than  1.4.  It  is  of  course  evident 
that  this  theoretical  value  is  not  actually  realized 
in  the  air  compressor  cycle,  because  certain  heat  inter- 
changes must  inevitably  take  place,  so  that  in  place  of 
adiabatic  compression  the  compression  line  shows 
slightly  decreasing  pressures  as  the  volume  decreases, 
i.e.,  decreasing  below  the  values  which  you  would  expect 
from  adiabatic  compression.  Nevertheless,  the  devia- 
tion from  the  true  adiabatic  compression  is  so  small 
that  most  engineers  consider  any  marked  deviation 
to  be  indicative  of  leakage  rather  than  of  thermal 
interchanges. 

In  case  the  compressor  forms  part  of  a  heating  engine 
the  increase  in  the  temperature  during  compression  is  an 
essential  feature,  and  every  effort  should  be  made  to  pro- 
vent  loss  of  temperature  through  external  radiation  and 
conduction,  but  if  the  temperature  at  which  the  heat 
is  delivered  is  of  no  importance  and  the  total  value  of 


AIR-COMPRESSORS  AND  AIR-MOTORS.  309 

the  operation  coasists  either  in  removing  heat  for 
refrigerative  purposes  or  in  storing  up  compressed 
fluids  for  future  power  purposes,  the  work  required 
for  adiabatic  compression  represents  a  real  loss,  because 
the  same  result  could  theoretically  be  accomplished 
by  isothermal  compression  to  the  upper  pressure  limit. 
Thus,  in  the  case  of  air-compressors  supplying  air  for 
air-motors  such  as  rock-drills,  etc.,  the  hot  air  delivered 
by  the  compressor  must,  during  its  transmission  through 
the  pipe-line,  be  reduced  in  temperature  to  that  of  the 
surrounding  atmosphere  and  therefore  the  air  occupies, 
at  the  pipe-line  pressure,  that  volume  which  it  would 
attain  provided  it  underwent  isothermal  compression 
from  the  initial  condition,  and  therefore  the  difference 
between  the  work  of  adiabatic  compression  and  the 
work  of  isothermal  compression  represents  useless 
expenditure  of  power. 

In  order  to  obtain  in  a  motor  power  equal  to  that 
required  for  compression,  it  would  be  necessary  to 
pre-heat  the  air  up  to  the  final  temperature  of  com- 
pression before  it  entered  the  motor.  Actually  without 
pre-heating  the  expansion  of  the  air  in  the  motor  cools 
the  temperature  of  the  exhaust  to  such  a  point  that 
the  water  entrained  in  the  atmosphere  frequently  freezes 
in  the  exhaust  ports. 

In  Fig.  108,  let  ab  and  ac  represent  isothermal  and 
frictionless  adiabatic  compression  respectively  from 
some  lower  pressure  pv  to  a  higher  pressure  pz.  If  the 


310        THE    TEMPERATURE-ENTROPY  DIAGRAM. 

temperature  of  the  cooling  water  is  the  same  as  that 
of  the  atmosphere,  the  minimum  possible  expenditure 
of  work  in  compressing  from  a  to  6  equals  that  under 


the  isothermal  ab,  or 


— .     If,   however,  the 


FIG.  108. 

compression  is  adiabatic  it  will  be  necessary  for  the 
delivered  air  to  contract  at  constant  pressure,  losing 
by  conduction  and  radiation  the  heat  cp(tc—tb).  That 


is,  the  work  T—  H  (  — 
K  —  1  L  \pa 


—1     is  performed  upon  the 


gas  during  compression,  as  shown  under  ac,  and  the 
work  pb(vc~vb)  during  contraction  in  the  storage  tubes 
or  coil,  as  shown  by  cb  in  the  ^-diagram.  The  wasted 
work  is  thus  shown  by  the  area  abc,  and  has  the 
value 


AIR-COMPRESSORS  AND   AIR-MOTORS.  311 

Jfc-l 


=  Acp[Tc-  To]  -ATa[<f>a  -  <pb]. 

In  the  TV-diagram  the  minimum  amount  of  heat 
rejected  is  shown  under  ab,  while  that  rejected  during 
contraction  at  constant  pressure  after  adiabatic  com- 
pression is  shown  under  cb,  the  heat  wasted  by  the 
latter  process  being  shown  by  acb.  If,  as  is  usually 
the  case,  the  compression  line  lies  somewhere  between 
these  two  extremes,  the  wasted  work  and  heat  will  be 
represented  by  some  such  area  as  adb. 

If  the  cooling  water  is  colder  than  the  atmosphere, 
it  is,  theoretically  at  least,  possible  to  reduce  the  neces- 
sary work  by  cooling  the  entering  gas  at  constant 
pressure  from  a  to  a',  compressing  isothermally  to  br, 
and  then  permitting  the  gas  to  warm  up  at  constant 
volume  by  taking  heat  from  the  atmosphere.  The 
work  performed  and  the  heat  rejected  during  com- 
pression are  represented  by  the  areas  under  a'b'  in  the 
two  diagrams,  and  the  work  saved  by  aa'b'b.  If  the 
compression  from  a'  is  along  the  adiabatic  aV,  the 
saving  over  that  along  ac  is  shown  by  the  area  aa'c'c. 

It  is  not  possible  to  cool  the  hot  gas  very  much  by 


312       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


jacketing,  but  the  waste  work  may  be  reduced  by 
dividing  the  compression  into  two  or  more  stages  and 
cooling  the  gas  in  intermediate  coolers  to  the  initial 
temperature. 

Thus  suppose  the  compression  to  follow  the  law 
pivln  =  p2v.in,  and  to  be  represented  by  the  curve  ac  in 
Fig.  109.  Instead  of  completing  the  compression  in  one 
cylinder,  stop  at  some  intermediate  pressure  px,  at  d, 


FIG.  109. 


and  cool  under  constant  pressure  to  e.  Continue  the 
compression  in  a  second  cylinder  along  ef,  and  finally 
cool  at  constant  pressure  along  /&.  The  wasted  work 
or  the  heat  ejected  is  no  longer  represented  by  the 
whole  of  abc,  but  by  the  two  portions  ode  and  efb; 
that  is,  the  work  or  heat  saved  by  compounding  is 
represented  by  the  area  cdef.  From  the  diagram  it  is 
at  once  evident  that  this  area  approaches  zero  as  px 
approaches  either  pt  or  p2,  and  that  there  is  some  inter- 
mediate position  which  gives  the  maximum  saving.  The 


AIR-COMPRESSORS  AND  AIR-MOTORS. 


313 


proper  value  of  px  is  easily  found  from  the  expression 
for  work 


This  expression  has  its  minimum  value  when 

n-l 

r>°. 

i.e.,  when  pt  :  px  =  px  '•  p^     °r    Px  =  ^pip2- 

Fig.  110  shows  similar  diagrams  for  a  three-stage 
compressor  with  intercoolers.  The  saving  thus  intro- 
duced is  shown  by  the  irregular-shaped  figure  defghc. 
This  again  varies  in  magnitude  with  the  values  of 


FIG.   110. 


p*  and  pa,  and  in  a  manner  similar  to  the  above  may 
be  shown  to  have  its  maximum  value  when 


Pi  '• 


or 


>Pv=Py  •  P2J 
and     py  =  V 


314       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Naturally  the  greater  part  of  the  saving  is  obtained 
by  the  use  of  one  extra  cylinder,  the  gain  from  successive 
extra  cylinders  becoming  correspondingly  smaller. 

Taylor  Hydraulic  Air-compressor.  —  To  make  the 
compression  line  coincide  with  the  isothermal  through- 
out, would  require  an  indefinite  number  of  stages, 
.  but  as  the  introduction  of  each  new  stage  increases 
the  total  friction  loss  of  the  compressor  the  number 
of  stages  actually  feasible  is  limited  to  three,  or  possibly 
four.  There  is,  however,  one  type  of  compressor 
essentially  simple  in  construction  which  realizes  almost 
perfectly  this  ideal  condition.  This  is  what  is  known 
as  the  Taylor  hydraulic  air-compressor,  consisting 
mainly  of  a  well,  sunk  at  some  point  where  a  drop  of 
water  is  available,  as  at  any  waterfall,  and  containing 
a  submerged  bell-shaped  chamber  which  is  connected 
through  the  top  by  a  pipe-line  leading  to  the  upper 
level  of  the  water,  this  pipe-line  extending  downward 
in  the  bell  part  way  from  the  top.  Water  from  the 
upper  level  flows  through  this  pipe  into  the  bell,  keeping 
the  lower  end  of  the  pipe  submerged. 

To  introduce  the  compressive  feature  there  is  a  series 
of  small  tubes  with  the  lower  ends  inserted  in  the  water 
entering  the  top  of  this  tube.  The  rush  of  the  water 
serves  to  draw  bubbles  of  air  down  through  these 
tubes,  which  bubbles  are  caught  in  the  descending 
water  and  carried  downward  into  the  bell.  Here  the 
stream  is  deflected  by  horizontal  aprons,  thus  giving 


AIR-COMPRESSORS   AND   AIR-MOTORS.  315 

the  air  a  chance  to  separate  from  the  water  by  gravity ; 
the  air  collecting  in  the  top  of  the  bell,  the  water  passing 
out  underneath  the  lower  edge  of  the  bell  up  around 
the  entire  mechanism  to  the  lower  level  of  the  fall. 
By  this  means  the  air  in  the  top  of  the  bell  is  subjected 
to  a  pressure  equal  to  the  weight  of  water  correspond- 
ing to  the  head,  measured  from  the  top  of  the  water 
in  the  submerged  bell  up  to  the  lower  level  of  the  water 
at  the  fall.  It  is  evident  that  the  energy  input  of  this 
compressor  is  equal  to  the  weight  of  water  passing 
through  the  tube,  multiplied  by  the  height  of  the  water- 
fall. The  useful  output,  on  the  other  hand,  is  repre- 
sented by  the  work  necessary  to  compress  the  actual 
quantity  of  air  delivered  isothermally  from  atmospheric 
pressure  to  the  pressure  prevailing  in  the  submerged 
bell.  Evidently  to  obtain  a  maximum  efficiency  the 
in-rush  of  air  should  be  so  regulated  that  the  maximum 
quantity  of  air  possible  should  be  carried  down  by  the 
water.  The  actual  loss  in  this  instrument  is  occasioned 
by  the  buoyancy  effect  due  to  gravity.  The  air  particles 
as  they  pass  down  through  the  tube  are  subjected  to 
ever  greater  pressures,  decrease  in  volume,  and  thereby 
assume  greater  density,  so  that  the  buoyancy  effect 
diminishes  as  they  descend  in  the  tube.  The  buoyancy 
effect,  however,  tends  to  maJce-the  bubbles  move  upward 
relatively  to  the  falling  water,  so  that  part  of  the  work 
already  accomplished  in  moving  them  downward  is 
wasted  by  this  retrograde  action. 


316        THE  TEMPERATURE-ENTROPY  DIAGRAM.  ' 

The  Influence  of  the  Clearance  Space. — Returning 
once  more  to  the  piston  compressor  it  is  next  necessary 
to  consider  the  effect  of  the  clearance  space.  As  we 
have  seen,  the  air  under  high  pressure,  stored  in  the 
clearance  space  at  the  beginning  of  the  stroke,  expands 
as  the  piston  moves  outward,  and  fills  a  certain  portion 
of  the  piston  displacement,  thereby  decreasing  the 
space  actually  available  for  new  air  during  the  succion 
stroke.  In  other  words,  the  actual  effective  displace- 
ment of  a  compressor  is  less  than  its  apparent  displace- 
ment, as  determined  from  its  dimensions  and  length 
of  stroke.  The  ratio  of  the  actual  displacement  to  the 
apparent  displacement  is  known  as  the  displacement 
efficiency  of  the  compressor.  Sometimes  the  actual 
air  delivered  per  cycle  divided  by  the  apparent  dis- 
placement is  also  called  displacement  efficiency,  but 
as  this  ratio  also  includes  the  effect  of  leakage  by  the 
piston  and  valves  its  use  should  be  avoided.  It  is 
better  to  have  separate  ratios:  one  defining  the  influence 
of  the  clearance,  the  other  the  effect  of  leakage  upon 
the  delivered  air.  Therefore,  in  calculating  the  size 
of  the  compressor  to  perform  a  stated  amount  of  work 
so  as  to  deliver  a  definite  volume  of  free  air,  the  piston 
displacement  calculated  directly  from  the  volume  of  the 
air  to  be  compressed  and 'delivered  must  be  divided  by 
this  displacement  efficiency  in  order  to  obtain  the  actual 
piston  displacement.  The  power  required  to  drive 
the  actual  compressor  (neglecting  friction  losses)  is 


AIR-COMPRESSORS   AND  AIR-MOTORS.  317 

the  same  as  the  power  that  would  be  required  to  drive 
a  compressor  having  no  clearance  space,  because,  as  the 
heat  interchange  is  small  between  air  and  cylinder 
walls,  the  expansion  and  compression  lines  are  practically 
the  same,  so  that  the  extra  work  required  to  compress 
this  clearance  air  is  returned  by  expansion  of  the  air 
during  the  suction  stroke. 

If  x  =  the  increase  in  volume  of  the  clearance  air  dur- 
ing expansion  from  delivery  (/?2)  to  suction  (pi)  pressure, 
and  n  =  the  exponent  of  the  expansion  curve,  while  Cl. 
and  P.D.  represent  the  volume  of  the  clearance  and 
piston  displacement  respectively,  there  exists  the 
simple  relation 


J^V-ilci. 
L\W 


whence 


From   this   it   follows   that    the   actual   displacement 
equals 

P.D.-X 


The  displacement  efficiency  thus  becomes  equal  to 


" 


318        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Determination  of  the  Clearance  of  an  Air-compressor 
from  the  Indicator  Card. — (a)  It  may  be  assumed 
without  much  error  that  the  values  of  the  exponent 


\ Atmospher 


Discharge  pressure 


\          Atmospheric  Lino 


FIG.  111. 

n  for  the  expansion  and  compression  curves  are  the 
same.  Let  these  curves  be  intersected  by  any  two 
straight  lines  parallel  to  the  atmospheric  line  in  the 
points  3  and  4  and  1  and  2,  respectively  (in  Fig. 
Ill  the  atmospheric  line  is  utilized),  so  that  ps  =  p\ 
and  p4  =  p2-  Determine  v\,  v2,  v3,  and  v4  in  per  cent. 
of  the  piston  displacement,  and  let  c  represent  the 
volume  of  the  unknown  clearance  in  per  cent,  of  the 
piston  displacement. 
Then 


and 


AIR-COMPRESSORS  AXD  AIR-MOTORS.  319 

whence 


or 

c — 


As  neither  p  nor  n  enter  into  the  value  of  c,  neither 
the  scale  of  the  indicator  spring  nor  the  actual  law 
of  expansion  and  compression  need  to  be  known. 

As  the  percentage  errors  are  much  larger  in  the  deter- 
mination of  v3  and  v4  than  in  v\  and  v2,  the  accuracy 
of  the  method  may  be  increased  by  assuming  a  prob- 
able value  for  the  clearance  and  adding  this  to  v\t 
v2,  v3,  and  v4.-  The  resulting  value  of  c  thus  obtained 
will  be  not  the  clearance  itself,  but  the  difference  be- 
tween the  actual  and  assumed  value,  and  may  be 
either  positive  or  negative  according  to  the  error  in 
the  assumption. 

(6)  When,  as  is  usually  the  case,  the  scale  of  the 
card  is  known  greater  accuracy  may  be  attained  by 
using  only  the  compression  line.  Intersect  the  curve 
at  pi  and  p2  =  %pi,  and  also  at  p3  and  p4  =  $p3.  Deter- 
mine the  corresponding  volumes  v\,  v2,  v3,  and  v^ 

Then 


and 


320       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


whence 


Actual  Air-compressor  Indicator  Cards. — Fig.  Ill 
shows  indicator  cards  from  the  two  cylinders  of  a 
small  single-stage  air-compressor  in  the  laboratories 
of  the  Institute.  In  the  lower  card  the  discharge- 
valve  was  functioning  properly,  in  the  upper  card  it 
opened  and  closed  spasmodically.  The  admission- 
ports  of  the  lower  card  offered  more  resistance  than 
those  in  the  upper  card,  as  shown  by  the  dropping 
suction  line. 

The  indicator  card  does  not  represent  a  closed  cycle, 
and  hence  if  projected  into  the  2*<£-plane  only  por- 
tions of  it  have  any  significance,  namely,  the  com- 
pression and  expansion  lines  and  the  enclosed  area. 
It  is,  however,  instructive  to  study  the  compression 
line,  as  this  gives  some  clue  to  the  effectiveness  of  the 
jacketing.  In  any  case,  the  compression  line  should  not 
deviate  much  from  the  adiabatic.  The  presence  of  a 
large  deviation  is  indicative  of  a  leak  instead  of  a 
heat  interchange. 

Various  Efficiencies. — As  we  have  seen,  part  of  the 
work  of  compression  is,  as  far  as  useful  output  is  con- 
cerned, wasted  energy,  so  that  it  has  become  customary 
to  consider  the  power  required  for  isothermal  compres- 
sion of  the  air  to  represent  the  useful  output  of  the 


AIR-COMPRESSORS  A\D  AIR-MOTORS.  321 

compressor  and  to  define  the  ratio  of  this  isothermal 
work  to  the  actual  indicated  work  as  the  air  efficiency, 
or  sometimes  as  compressor  efficiency.  In  the  case 
of  the  ideal  engine  the  air  efficiency  would  equal  the 
ratio  of  the  isothermal  work  to  the  adiabatic  work 
for  the  proper  number  of  stages. 

Usually  such  compressors  are  driven  either  by  direct 
connected  motors  or  steam  engines,  or  are  belted  to 
such  motors.  There  is  usually  no  attempt  made  to 
separate  the  friction  losses  in  such  combined  units 
of  the  motor  and  of  the  compressor,  but  the  total 
friction  loss  is  obtained  by  determining  the  difference 
in  the  indicated  power  of  the  steam  cylinders  and  the 
air  cylinders,  and  the  mechanical  efficiency  of  such  a 
unit  is  defined  as  the  ratio  of  the  indicated  air  power 
to  the  indicated  steam  power.  Of  course  in  the  case 
of  an  electric  motor  operating  such  a  compressor  the 
efficiency  of  the  unit  would  be  represented  by  the  ratio 
of  the  indicated  air  power  to  the  electrical  input  at 
the  terminals  of  the  motor. 

The  values  of  these  various  efficiencies  actually 
attained  in  practice  vary  between  very  wide  limits  due 
to  different  forms  of  compressors  and  the  varying 
degrees  of  workmanship.  Thus  the  displacement  effici- 
ency ranges  between  78  per  cent  and  94  per  cent,  i.e., 
the  effect  of  the  clearance  is  to  decrease  piston  displace- 
ment by  22  to  6  per  cent  of  its  actual  value. 
The  air  efficiency  or  the  compression  efficiency  is 


322       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

influenced  by  the  action  of  the  admission  and  delivery 
ports  and  also  by  the  heat  interchanges.  The  heat 
losses  are  greater  in  compound  than  simple  compressors 
because  the  air  comes  in  contact  with  two  cylinders 
instead  of  one.  In  certain  of  the  better  types  of  com- 
pressors the  valves  are  not  operated  by  difference  of 
pressure,  but  by  mechanical  means.  In  these  latter 
compressors  the  throttling  loss  during  admission  and 
exhaust  is  much  diminished.  Numerically  the  efficiency 
of  compression  varies  from  about  50  to  90  per  cent.  On 
page  371,  Peabody,  there  is  a  table  giving  the  values 
found  in  tests  on  a  variety  of  compressors. 

The  mechanical  efficiency  of  the  unit  is  ordinarily 
in  the  neighborhood  of  85  per  cent,  but  occasionally 
reaches  higher  values  as,  for  example,  page  372,  Pea- 
body,  mechanical  efficiency  for  blowing  engines  at 
Creusot  is  quoted  at  92  per  cent.  From  the  values 
given  for  the  air  and  mechanical  efficiencies  it  is 
evident  that  the  total  efficiency  of  the  compressor  unit, 
which  may  be  defined  as  the  isothermal  work  of  com- 
pression divided  by  the  indicated  power  of  the  steam 
cylinders,  ranges  from  40  to  75  per  cent. 

It  is  interesting  to  compare  these  values  with  those 
obtained  with  hydraulic  compressors.  Thus  reports 
of  tests  made  upon  the  Taylor  Hydraulic  Compressor 
show  that  the  ratio  of  the  isothermal  work  on  the  air 
to  the  water  work  ranges  from  60  to  70  per  cent. 
This  corresponds  to  the  total  efficiency  of  a  steam- 


AIR-COMPRESSORS  AND  AIR-MOTORS  323 

driven  air- compressor  unit  and  shows  that  the  results 
obtained  for  Taylor  compressors  always  exceed  the 
minimum  values  of  the  steam-driven  compressor,  and 
usually  the  values  very  nearly  equal  to  the  maximum 
one  obtained  with  steam-driven  compressors.  It  is 
of  course  possible  to  use  water  power  to  operate  a 
water  turbine  and  to  connect  this  turbine  to  a  piston 
air-compressor.  Such  water  turbines  range  in  efficiency 
from  20  to  80  per  cent.  If  for  a  basis  of  comparison 
we  take  this  maximum  efficiency  of  the  water  turbine 
as  80  per  cent  and  multiply  it  by  a  high  value  for  the 
efficiency  of  compression  (say  80  per  cent)  this  would 
give  as  the  total  efficiency  of  a  water  turbine  air-com- 
pressor unit  a  value  of  64  per  cent,  which  is  readily 
exceeded  by  the  Taylor  compressor. 

Applications  of  Compressed  Air. — Compressed  air  is 
used  in  almost  all  branches  of  mechanical  and  chemical 
engineering,  the  scope  of  this  utilization  being  indicated 
briefly  by  the  following  headings:  (1)  Mechanical  Draft, 
Cooling  Towers,  Ash  and  Coal  Conveyors,  (2)  Heating 
and  Ventilating,  (3)  Air  Drying,  Lumber  Drying  Kilns, 
etc.,  (4)  Steel  Production  and  Manipulation,  as  Forges, 
Hammers,  Drills,  Riveters,  etc.,  (5)  Air  Motors  and  Rock 
Drills,  (6)  Exhausters  to  Promote  Increased  Speed  of 
Evaporation,  etc.,  (7)  Pneumatic  Conveying  Systems, 
Planing  Mill  Exhausters,  (8)  Sand  Blast  for  Cleaning 
Iron,  Stone,  etc.,  (9)  Paint  Blast  for  Painting  Bridges, 
Freight  Cars,  etc.,  (10)  Pumps  of  both  Displacement  and 


324        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Expansion  type,  of  which  latter  the  Pohle*  air  lift  is  a 
good  example,  and  the  combination  of  both  methods, 
such  as  the  Starrett  air  pump. 

Types  of  Compressors. — These  various  applications  call 
for  air  pressures  varying  from  a  fraction  of  an  ounce  to  the 
square  inch  up  to'possibly  ten  pounds  to  the  square  inch. 
In  the  case  of  the  air-motors  and  for  the  production  of 
liquid  air,  and  some  few  other  devices,  much  higher 
pressures  extending  up  to  the  hundred  or  thousand,  or 
several  thousands,  of  pounds  to  the  square  inch  are 
required.  To  produce  air  of  such  varying  pressures 
a  variety  of  mechanisms  has  been  devised.  For  low 
pressures  extending"  up  to  one  ounce  or  less  per  inch,  a 
disk  or  propeller  fan  running  at  slow  speed  \vill  handle 
large  volumes  of  air  at  small  cost.  For  pressures  up  to  one 
pound  per  square  inch  centrifugal  fan  blowers  consisting 
of  steel  wheels  running  at  high  speeds  in  conical  casings 
will  handle  large  volumes  of  air  at  comparatively  small 
cost.  In  both  these  types  of  fans  there  is  always  the 
possibility  for  air  to  leak  back  from  the  delivery  side  to 
the  suction  side  through  the  clearance  space,  so  that  for 
higher  pressures  some  more  positive  action  or  mechan- 
ism must  be  utilized.  Thus  for  pressures  ranging  to  ten 
pounds  we  have  a  rotary  type  of  blower  or  exhauster 
engine  which  consists  essentially  of  two  geared  wheels 
so  intermeshing  that  on  the  upward  stroke  or  motion  air 
is  carried  forward  between  the  teeth,  but  on  the  down- 
ward stroke  the  teeth  intermeshing  prevent  the  escape 


AIR-COMPRESSORS  AND   AIR-MOTORS.  325 

of  the  air,  so  that  it  is  delivered  into  the  discharge  pipe. 
These  blowers  handle  capacities  ranging  anywhere  from 
five  cubic  feet  to  over  15,000  cubic  feet  per  minute 

The  piston  compressor  is  designed  to  deliver  air  at  any 
pressure  up  to  three  or  four  thousand  pounds  per  square 
inch.  The  Taylor  hydraulic  compressor  occupies  a 
unique  position  in  that  it  attains  most  closely  to  isother- 
mal compression.  It  is  capable  of  supplying  large  vol- 
umes of  air  at  ordinary  commercial  pressures. 

Use  of  Compressed  Air  for  Pumping. — As  a  simple, 
economical,  and  reliable  power  for  pumping  purposes, 
compressed  air  for  many  applications  cannot  be  sur- 
passed. The  ordinary  types  of  air  pumps  fall  under  two 
general  headings,  (1)  those  using  the  pressure  of  the  air 
alone  to  displace  water  from  a  suitable  chamber,  (2) 
those  using  the  expansive  force  of  the  air  by  interming- 
ling it  with  the  column  of  ascending  water,  as  in  the 
Pohle  air  lift.  There  is  a  third  type  which  is  really  a 
combination  of  these  two,  a  good  illustration  being  the 
Starrett  pump. 

In  the  displacement  type  of  pumps  the  compressed  air 
is  led  into  the  top  of  the  pump  chamber  and  the  water 
is  forced  out  through  a  pipe  from  the  bottom.  When 
the  water  is  completely  displaced  the  supply  of  air  is 
shut  off  and  the  air  in  the  chamber  is  either  exhausted 
to  the  atmosphere  or  in  some  types  is  fed  back  during 
the  suction  stroke  of  the  compressor.  The  weight  of 
water  in  the  delivery  pipe  closes  the  check  valve  and 


326        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

prevents  water  flowing  back  into  the  chamber.  The 
decrease  in  pressure  in  the  chamber,  however,  permits 
water  to  enter  through  an  admission  pipe  either  under 
the  influence  of  gravity  (in  case  the  chamber  is  sub- 
merged) or  under  the  influence  of  atmospheric  pressure 
in  case  the  air  pressure  is  reduced  by  drawing  it  into  a 
compressor. 

Ordinarily  such  pumps  are  made  with  two  cylinders, 
so  that  one  is  discharging  while  the  other  is  filling,  thus 
giving  a  fairly  constant  delivery  of  the  water.  Such  a 
method  of  pumping  utilizes  simply  the  direct  pressure  of 
the  air  and  fails  to  utilize  any  of  its  expansive  force  or 
internal  energy.  The  economy  obtained  by  such  a 
method  must  be  comparable  with  that  obtained  in 
direct-acting  steam  pumps  where  the  steam  is  used  non- 
expansively. 

The  Pohle"  Air  Lift  consists  of  two  tubes,  one  a  large 
tube  with  a  bell-shaped  enlargement  at  the  bottom,  sub- 
merged to  a  considerable  extent  in  the  well  from  which 
water  is  to  be  pumped.  The  other  is  a  small  pipe  for 
supplying  the  compressed  air  which  it  delivers  under- 
neath the  bell-shaped  opening  of  the  larger  tube.  The 
result  is  that  the  compressed  air  rising  upward  under  the 
influence  of  gravity  tends  to  carry  with  it  slugs  of  water. 
We  have  thus  in  this  tube  a  rising  column  of  alternating 
layers  of  water  and  air,  or  a  column  of  water  interspersed 
with  numerous  air  bubbles.  This  results  in  a  dimin- 
ished de'nsity  of  the  combined  mass,  so  that  the  weight 


AIR-COMPRESSORS  AND  AIR-MOTORS.  327 

of  water  outside  the  pipe  will  balance  a  considerably 
higher  head  of  this  mixture  inside  the  tube.  The  greater 
the  height  to  which  the  water  must  be  raised  the  less  the 
density  of  the  mixture  in  the  tube.  It  is  to  be  noticed 
that  this  Pohle  air  lift  is  simply  .the  reverse  of  the  Taylor 
air-compressor.  Thus,  as  the  bubbles  rise  in  the  pipe, 
they  are  subjected  to  ever  decreasing  pressure,  namely, 
the  weight  of  the  water  above  them,  and  therefore  ex- 
pand. The  tendency  to  cool,  due  to  expansion,  is  offset 
by  the  rapid  interchange  of  heat  from  the  water  to  the 
air,  due  to  the  intimate  mixture.  The  expansion  of  the 
air  is  therefore  practically  isothermal,  and  the  work 
developed  is  thus  the  maximum  which  could  possibly 
be  attained  without  the  use  of  an  external  source  of 
heat.  The  sources  of  loss  in  this  pump  are  (1),  in  the 
absorption  of  air  by  water  so  that  the  whole  of  the  air 
supplied  cannot  be  utilized  for  pumping  purposes,  and 
(2),  in  the  fact  that  water  from  the  bottom  of  one  slug 
tends  to  leak  downward  to  the  top  of  the  one  following, 
so  that  there  is  a  continual  backward  flow  of  water  down 
the  pipe,  i.e.,  water  once  lifted  falls  back  and  must  be 
pumped  up  again,  thereby  wasting  part  of  the  power  of 
the  air.  Experiments  on  certain  Pohle  air  lifts  have 
shown  an  efficiency  of  about  50  per  cent. 

The  Starrett  Air  Pump  utilizes  a  combination  of  the 
displacement  pump  and  the  Pohle  air  lift.  Essentially 
it  consists  of  two  displacement  chambers  connected  to  a 
common  discharge  pipe  and  supplied  through  an  auto- 


328        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

matic  valve  from  a  common  air  pipe.  One  cylinder  is 
filling  under  the  influence  of  gravity,  while  the  other  is 
being  emptied  by  means  of  the  compressed  air.  By 
means  of  a  rather  complicated  valve  action  a  portion  of 
the  air  used  for  displacement  escapes  into  the  column  of 
ascending  water,  the  rest  of  it  being  discharged  directly 
to  the  air.  The  valve  also  bleeds  a  certain  quantity  of 
compressed  air  directly  into  the  discharge  pipe.  It  is 
evident  that  all  the  air  discharged  to  the  atmosphere 
escapes  with  its  internal  energy  unutilized,  and  there- 
fore fails  to  give  as  good  economy  as  the  air  which  passes 
into  the  delivery  pipe.  Theoretically,  at  least,  such  a 
pump  cannot  have  as  high  a  thermal  efficiency  as  the 
simple  Pohle  air  lift.  It  has,  however,  certain  other 
advantages  which  in  many  installations  would  make  it 
more  advantageous  than  the  Pohle  air  lift.  Thus  the 
Pohle  air  lift  requires  that  the  pipe  be  submerged  to  such 
a  distance  beneath  the  level  of  the  water  that  the  weight 
of  the  water  outside  the  tube  shall  overbalance  the 
longer  column  of  water  and  air  inside  the  tube.  In 
other  words,  such  a  method  could  not  be  utilized  unless 
the  well  was  sunk  far  below  the  level  of  the  water.  The 
Starrett  pump,  on  the  other  hand,  may  be  placed  just 
below  the  surface  of  the  water,  as  the  water  flows  into  it 
under  the  influence  of  gravity,  and  therefore  does  not 
need  such  a  deep  excavation.  It  is  quite  possible  that 
the  decreased  cost  of  installation  might  offset  any  ther- 
mal advantages  inherent  in  the  Pohle  air  lift.  Tests 


AIR-COMPRESSORS  A\'D  AIR-MOTORS.  329 

made  upon  the  Starrett  pump  installed  in  the  labora- 
tories of  the  Institute  have  shown  efficiencies  of  about 
fifty  per  cent,  which  for  some  strange  reason  seem  to  be 
higher  than  the  efficiencies  which  are  quoted  for  the 
Pohle  air  lift. 

There  is  one  interesting  feature  common  to  both  the 
Pohle  and  the  Starrett  pump,  and  that  is  that  during 
the  practically  isothermal  expansion  of  the  air  in  the 
delivery  pipe  heat  is  flowing  from  the  water  to  the  air, 
therefore  increasing  the  work  developed  by  the  air;  or, 
in  other  words,  part  of  the  heat  energy  of  the  water  is 
being  utilized  to  help  pump  it. 

Compressed  Air  Used  as  a  Source  of  Power. — Return- 
ing to  the  first  general  case  where  the  air  is  used  for 
power,  it  is  necessary  first  of  all  to  discuss  the  influence 
of  the  pipe  line  which  conducts  the  air  from  the  com- 
pressor to  the  machine  to  be  operated  by  the  compressed 
air,  such  as  a  rock-drill,  a  penumatic  riveter,  a  com- 
pressed-air motor,  etc.  The  temperature  of  the  pipe 
may  be  considered  as  equal  to  that  of  the  surrounding 
atmosphere  and  hence  constant.  The  heat  generated 
by  friction  is  thus  at  once  dissipated  by  conduction, 
and  there  thus  results  an  isothermal  drop  of  pressure 
and  increase  of  volume. 

Thus,  if  abcde,  Fig.  112,  represent  the  passage  of  the 
air  through  the  compressor,  ef  shows  the  lo?s  experienced 
by  the  air  in  flowing  from  the  compreasor  to  the  engine. 
Suppose  the  air  at  /to  expand  adiabatically  in  the  motor 


330         THE  TEMPERATURE-ENTROPY  DIAGRAM. 


down  to  back  pressure,  the  amount  of  work  performed 
will  be  equal  to  the  area  under  fg  in  the  pv-diagram. 
The  exhaust  air  is  now  warmed  to  the  initial  tempera- 
ture along  the  constant-pressure  curve  ga,  thus  per- 


FIG.  112. 

forming  upon  the  atmosphere  the  work  under  ga  in 
the  pv-plane,  and  receives  from  the  atmosphere  the 
heat  under  ga  in  the  T</>-plane.  The  maximum  amount 
of  work  could  be  obtained  from  such  an  engine  if  the 
expansion  were  along  the  isothermal  fa.  This  can  be 
partially  attained  by  jacketing  with  water  at  atmos- 
pheric temperature,  so  that  the  actual  expansion-curve 
lies  somewhere  between  these  two  limiting  cases,  as  at 
}g/.  A  further  gain  could  be  made  by  compounding 
the  engine  and  heating  the  air  up  to  atmospheric  tem- 
perature in  the  intermediate  receiver,  as  indicated  by 
fhkla. 

Effect  of  Pre-heating  the  Air. — The  amount  of  vrork 
theoretically  obtainable  from  a  pound  of  air  in  a  non- 
conducting motor  is  given  by  the  formula : 


AIR-COMPRESSORS  AND   AIR-MOTORS.  331 


0.405 
,1.405 


pjO-2883 


and  is  thus  shown  to  be  directly  proportional  to  its  ini- 
tial temperature.  In  the  operation  of  motors  by  com- 
pressed air  it  is  thus  possible  to  bring  about  a  consid- 
erable saving  in  the  amount  of  air  required  as  well  as 
an  improvement  in  the  thermal  efficiency  in  any  given 
case  by  heating  the  air. 

This  heating,  or  pre-heating,  as  it  is  generally  termed, 
may  be  accomplished  either  by : 

1.  The  direct  application  of  heat  in  some  form  of  com- 
bustion heater,  or 

2.  The  admixture  of  steam. 

This  latter  method  may  operatively  be  effected  by  one 
of  two  methods,  viz.,  by  blowing  steam  into  the  air  main 
or  by  passing  the  air  through  a  boiler,  in  which  latter 
operation  it  becomes  heated  to  the  temperature  of  the 
steam. 


332        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

Air-motor  Economy.— In  air-motor  work  it  is  custom- 
ary tor  quote  the  consumption  in  terms  of  cubic  feet  of 
free  air  per  indicated  horse-power  per  hour.  This  result 
may  be  derived  from  the  preceding  equation  as  follows : 

33000X00 
Lbs.  of  air  per  I.H.F.  hour  =  — 


1072 


Cubic  feet  free  air  per  I.H.P.  hour  =  lbs.  air  X  vol.  of 
1  Ib.  air  at  atmospheric  temperature 


1 079 


270.27\ 


The  maximum  air  consumption  evidently  occurs 
when  the  air  in  the  pipe-line  has  been  cooled  to  atmos- 
pheric temperature,  and  is  equal  to 


'         cu.ft.  per  I.H.P.  hour. 


AIR-COMPRESSORS  AND  AIR-MOTORS. 


333 


The  cubic  feet  for  any  amount  of  pre-heating  may  then 

T 

be  obtained  by  multiplying  this  result  by  •=£ 

1 1 

Air-motor  Efficiencies. — Assuming  the  air  to  be  sup- 
plied by  a  single-stage  compressor,  and  that  before 
entering  the  motor  it  is  cooled  to  atmospheric  temper- 


FIG.  113. 
ature,then  the  work  of  adiabatic  compression  would  be 


(Fig.  113): 


so  that  the  efficiency  of  the  compressor-motor  unit  would 
be: 


If  the  compressor  is  driven  by  a  steam  engine,  the  effi- 
ciency of  the  motor  is  equal  to  the  combined  efficiencies 
of  boiler,  Rankine  cycle,  cylinder,  mechanical  efficiency 
of  engine,  compressor  unit,  efficiency  of  motor  compres- 
sor, or 


334       THE  TEMPERATURE-ENTROPY  DIAGRAM. 
thermal  efficiency  of  motor  = 

^boiler  '  r/Rankine  '  ^cylinder  '  ^mechanical '  ^compressor  '  r/'motor  • 

The  value  of  this  thermal  efficiency  lies  well  inside  ten 
per  cent. 

Influence  of  Pressure  and  Temperature  upon  the  Indi- 
cated Work. — The  expression  for  the  work  developed  in 
an  air  motor: 


and  the  corresponding  expression  for  the  work  required 
to  compress  air : 


may  be  transformed  the  one  into  the  other  by  making 
use  of  the  expression  for  an  adiabatic  expansion  in  terms 
of  temperature  and  pressure  : 


IZL*  i-fc 

*~  = 


These  must  of  course  be  of  the  same  magnitude,  as  they 
represent  the  same  cycle  of  operations  simply  in  a  re- 
versed direction.  In  both  formula?  the  subscript  1 
applies  to  the  conditions  at  the  upper  pressure  level, 
and  subscript  2  the  conditions  at  the  lower  pressure 
level.  These  different  forms  for  the  same  quantity  are 


AIR-COMPRESSORS  AND  AIR-MOTORS.          335 

used  because  ordinarily  one  knows  the  initial  condition 
of  the  air  supplied  to  a  motor  and  would  not  be  liable 
to  know  the  exhaust  temperature,  and  similarly,  one 
knows  the  condition  of  the  air  supplied  to  the  compres- 
sor and  would  not  be  liable  to  know  its  exhaust  temper- 
ature. 

A  comparison  of  these  two  formulae  makes  evident 
the  fact  that  the  amount  of  work  produced  or  consumed 
is  absolutely  independent  of  the  actual  values  of  the 
upper  arid  lower  pressures,  but  is  dependent  solely  upon 
the  ratio  of  these  values.  Thus  the  amount  of  work 
required  to  compress  a  pound  of  air  from  1  Ib.  to  15  Ibs. 
pressure  is  equal  to  that  which  would  be  required  to 
compress  it,  say  from  15  Ibs.  to  225  Ibs.  Similarly  the 
work  required  to  compress  a  pound  of  air  from  .01  of  a 
pound  per  sq.  in.  to  one  pound  per  sq.  in.  is  as  great  as 
the  work  required  to  compress  it  from,  say  10  Ibs.  per 
sq.  in.  to  1000  Ibs.  per  sq.  in.  On  the  other  hand  the 
amount  of  work  obtained  or  absorbed  per  pound  of  air 
is  directly  proportional  to  the  initial  temperature  of  the 
charge  supplied  to  the  motor  or  to  the  compressor. 

If  therefore  it  is  essential  and  necessary  to  water-cooi 
the  cylinders  of  an  ordinary  air-compressor  which  is 
compressing  from  atmospheric  pressure  to  two  or  three 
hundred  pounds,  it  is  just  as  necessary  to  cool  the  cylin- 
ders of  the  dry  air  pump,  whose  function  it  is  to  take  the 
'urn*  pressure  air  from  a  vacuum  chamber  and  discharge 
it  into  the  atmosphere.  This  is  clearly  demonstrated  in 


336        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

the  temperature-entropy  plane  from  the  fact  that  the 
constant  pressure  cur.ves  all  possess  the  same  form  and 
are  simply  displaced  from  one  another  parallel  to  the 
entropy  axis,  the  displacements  being  proportional  to  the 
logarithms  of  the  pressure.  Thus  the  pressure  curves 
for  1000,  100,  10,  1,  Vio,  Vioo  Ibs.,  etc.,  would  form  a 


FIG.  114. 

series  of  curves  separated  by  equal  distarces  (Fig.  114). 
Therefore  the  work  of  compression  from  any  one  curve 
to  the  succeeding  higher  curve  would  be  the  same  as 
.hat  from  any  other  curve  to  its  succeeding  curve; 
assuming  the  compression  in  all  cases  to  start  at  the 
.^ame  temperature,  the  increase  in  temperature  would 
be  the  same  in  all  cases,  and  finally  the  heat  absorbed 
by  the  cooling  water  must  also  be  the  same  in  all  cases. 


CHAPTER  XVII. 
DISCUSSION  OF  REFRIGERATING   PROCESSES. 

Refrigerative  Units. — Various  branches  of  engineering 
have  their  own  units,  which  in  a  sense  are  historical 
records  of  the  functions  formerly  exercised  by  other 
devices  or  systems,  and  which  have  been  usurped  by 
these  particular  branches  of  engineering.  Thus  in  power 
development  the  name  of  the  horse  is  preserved  in  horse- 
power, Pferdstaerke,  cheval  vapeur,  etc.,  which  is  the 
unit  for  all  steam-  and  gas-engine  measurements ;  even 
in  boiler  rating  the  name  still  sticks,  although  its  numer- 
ical value  has  long  ceased  to  possess  any  relation  with 
its  origin. 

In  refrigeration  engineering  it  is  not  the  horse  but 
natural  ice  which  is  being  replaced.  As  heat  conduction 
is  a  continuous  operation,  the  refrigerating  agent  must 
operate  twenty-four  hours  a  day,  so  that  when  we  fill  our 
refrigerators  with  ice  the  supply  must  be  sufficient  to 
last  the  time  interval  from  one  filling  to  the  next.  In 
popular  or  commercial  parlance  the  refrigerating  needs 
of  any  plant  would  be  represented  by  the  rate  of  melt- 
ing of  the  ice,  as  so  many  units  of  weight  of  ice  per  day. 
It  is  but  natural,  then,  to  find  the  capacity  of  mechanical 

refrigerating  plants  expressed  as  the  equivalent  of  so 

337 


338        THE  TEMPERATURE-ENTROPY  DIAGRAM, 

much  ice  melted  daily.  The  unit  of  refrigeration  adopted 
by  the  A.S.M.E.  is  the  heat  absorbing  capacity  due  to  the 
melting  of  one  ton  of  ice  in  twenty-four  hours.  As  the 
latent  heat  of  fusion  of  ice  is  144  B.T.U.  per  Ib.  this 
unit  corresponds  to  the  absorption  of  288,000  B.T.U. 
per  24  hours,  or  12,000  B.T.U.  per  hour,  or  200  B.T.U. 
per  minute.  A  report  of  the  committee  to  the  society 
may  be  found  in  the  records  of  the  A.S.M.E.  in 
Vol.  28,  year  1906,  page  1249. 

The  output  of  a  refrigerating  plant  is  expressed  in 
various  ways,  according  to  the  feature  it  is  desired  to 
accentuate.  Thus  to  the  purchaser  or  user,  the  chief  item 
of  interest  is  the  ice  equivalent,  so  the  output  is  quoted  in 
tons  of  ice  melted  in  twenty-four  hours.  For  the  power 
engineer  the  important  question  is  the  horse-power  re- 
quired to  produce  a  certain  number  of  units  of  refrig- 
eration, so  that  the  output  is  usually  quoted  in  B.T.U. 
refrigeration  per  steam  horse-power  per  minute.  Finally, 
from  the  standpoint  of  cost,  reference  must  eventually 
be  made  to  the  coal  pile,  so  that  the  rcfrigerative  effect 
is  also  quoted  in  terms  of  pounds  of  ice  melted  per 
pound  of  coal  burned. 

Refrigerative  Systems.— Refrigerating-  plants  group 
naturally  under  one  of  three  headings : 

1.  Those  using  air  with  either  the  natural  or  dense 
air  system. 

2.  Those  using  some  volatile  Substance,  as  ammonia, 
with  a  compressor. . 


DISCUSSION  OF  REFRIGERATING   PROCESSES.  339 

3.  Those  using  some  volatile  substance,  as  ammonia, 
with  some  absorbent,  as  water,  with  its  necessary  ad- 
juncts, absorber,  generator,  analyser,  rectifier,  inter- 
changer. 

Air  Refrigeration.— Air  refrigeration  at  the  present 
time  is  used  chiefly  under  such  circumstances  as  make 
the  use  of  ammonia  refrigeration  impossible  or  inexpe- 
dient, the  advantage  of  this  method  being  that  leakage 
of  the  refrigerative  fluid  causes  no  inconvenience.  The 
disadvantage  lies  in  the  small  heat  capacity  of  the  air, 
which  is  only  0.2375  B.T.U.  per  pound  per  degree  dif- 
ference of  temperature,  so  that  large  volumes  of  air 
have  to  be  handled  by  the  compressor  and  the  rest  of 
the  system  in  order  to  affect  any  appreciable  refriger- 
ation. The  apparatus  used  consists  of  several  members, 
each  possessing  the  peculiarities  and  performing  the 
functions  described  in  the  following.  A  compressor 
which  may  if  desirable  be  two-staged,  the  function  of 
which  is  to  increase  the  pressure  of  the  air  a  consider- 
able amount.  The  involved  increase  of  temperature 
must  next  be  eliminated  by  carrying  the  delivery  pipe 
of  the  compressor  through  a  suitable  water  bath,  so 
that  the  air  is  restored  to  atmospheric  temperature. 
The  only  way  in  which  the  temperature  of  the  air  can 
lie  diminished  other  than  by  direct  radiation  to  some 
colder  body,  is  to  utilize  the  internal  energy  of  the  air 
in  the  performance  of  work.  This  is  most  effectively 
accomplished  by  expanding  the  air  in  a  motor  or  expan- 


340        THE   TEMPERATURE-ENTROPY  DIAGRAM. 

dor  which  ordinarily  is  either  direct  connected  to  the 
compressor  piston  rod  or  indirectly  through  a  common 
shaft  and  thus  serves  to  help  drive  the  compressor.  The 
difference  between  the  power  absorbed  by  the  compres- 
sor and  that  developed  by  the  motor  must  be  supplied 
from  some  external  source.  Because  of  the  practically 
adiabatic  expansion  in  the  motor  cylinder  the  final  tem- 
perature of  the  air  is  expressed  by  the  formula 


/PatmA 
\        2    I 


i-k 

k 


i.e.,  the  air  during  expansion  undergoes  the  same  frac- 
tional diminution  in  temperature  as  its  increase  during 
compression.  The  air  leaving  the  compressor  is  there- 
fore much  colder  than  the  atmosphere  and  may  be  used 
as  a  refrigerative  agent.  The  rest  of  the  system  consists 
of  a  pipe  conducting  the  cold  air  from  the  exhaust  of 
the  motor  to  the  refrigerating  rooms  or  brine  tank,  etc. 
As  the  air  circulates  through  these  various  chambers 
heat  flows  into  it  from  the  surrounding  hotter  atmo- 
sphere and  it  therefore  serves  to  cool  these  respective 
chambers.  Eventually  the  air  will  regain  atmospheric 
temperature,  so  that  the  theoretical  refrigerative  effect 
is  equal  to  0.2375  X(7Tatm.-772).  Actually,  however, 
the  full  value  of  this  refrigerative  effect  cannot  be  real- 
ized as  the  latter  portion  of  the  heat  is  received  at  tem- 
peratures greater  than  that  of  the  desired  refrigeration, 
so  that  this  would  be  transferred  through  the  piping 


DISCUSSION  OF  REFRIGERATING  PROCESSES.     341 

after  the  air  left  the  rcfrigerative .  chamber  and  is  on 
its  return  to  the  compressor.  Ordinarily,  therefore,  this 
final  section  of  the  pipe  is  omitted  and  the  air  is  wasted, 
a  new  supply  being  drawn  in  at  the  compressor.  In  air 
refrigerating  plants  the  work  performed  during  expan- 
sion is  no  longer  tho  main  object,  as  in  the  case  just 


discussed,  but  is  simply  a  means  of  obtaining  the  desired 
end,  viz.,  a  sufficient  drop  in  the  temperature  of  the  air. 
The  essential  parts  of  such  a  system  arc  shown  in  Fig. 
115.  The  compressor  A  takes  its  supply  of  air  from 
the  atmosphere  and  discharges  the  compressed  air  into 
the  cooling  coil  B.  where  temperature  and  volume  are 
both  decreased  at  constant  pressure.  The  cold  air  now 


342       THE   TEMPERATURE-ENTROPY   DIAGRAM. 


passes  into  C,  and  in  expanding  helps  to  operate  the 
compressor.  The  expanded  air  is  delivered  at  low  tem- 
perature and  atmospheric  pressure  to  the  refrigerator 
room,  and  as  it  passes  through  D  its  temperature  in- 
creases and  reaches  that  of  D  by  the  time  it  leaves  at 
the  right.  The  cycle  of  the  air  is  completed  by  warming 
to  the  initial  atmospheric  temperature  outside  of  the 


o  ,'/ 


FIG.  116. 

refrigerator.  The  expansion  in  C  being  used  to  attain 
low  temperature  instead  of  work,  care  is  taken  not  to 
heat  the  air  during  expansion,  so  that  the  expansion 
may  be  as  nearly  adiabatic  as  possible. 

In  Fig.  116  let  anb  represent  the  passage  of  the  air 
through  a  two-stage  compressor  A  and  the  cooling  tank 
B.  During  the  expansion  be,  in  the  working  cylinder  C, 
the  temperature  of  the  air  drops  below  that  maintained 
in  the  refrigerator  Tr,  the  air  being  delivered  at  pressure 


DISCUSSION  OF  REFRIGERATING  PROCESSES.    343 

p(l.  As  the  temperature  of  the  air  increases  along  the 
constant-pressure  curve  ca,  it  extracts  from  the  refrig- 
erator the  heat  under  the  curve  cd,  in  the  T^-plane 
The  heat  under  da,  necessary  to  complete  the  cycle,  is 
obtained  from  the  atmosphere.  That  is,  the  refrigera- 
ting effect  cdef  is  attained  by  the  expenditure  of  the 
work  abcda.  If  the  compressor  has  but  one  stage,  amb, 
the  efficiency  will  be  correspondingly  less. 

When  we  realize  that  a  pound  of  air  at  ordinary  at- 
mospheric conditions  occupies  about  13  cu.ft.  and  for  a 
range  of  temperature  of,  say  100  degrees,  has  a  refriger- 
ative  capacity  of  approximately  24  B.T.U.,  we  see  that- 
to  accomplish  any  appreciable  amount  of  refrigeration 
a  large  number  of  pounds  of  ajr  must  be  circulated  and 
therefore  the  volume  of  air  to  be  handled  is  practically 
prohibitive.  It  is  at  this  point  that  advantage  is  taken 
of  the  fact  previously  mentioned  that  the  work  of  com- 
pression is  independent  of  the  actual  pressures,  and  only 
dependent  upon  the  ratio  of  pressures  in  what  is  known 
as  the  Allen  Dense  Air  Machine.  In  this  mechanism 
the  air  is  never  expanded  down  to  atmospheric  pressure, 
but  reaches,  as  it  exhausts  from  the  motor,  a  pressure  of 
about  five  atmospheres,  or  roughly  00  Ibs.  gage,  and 
when  discharged  from  the  compressor  is  ordinarily 
about  15|  atmospheres,  or  roughly  210  Ibs.  gage. 

Assuming  that  the  air  entering  the  compressor  is 
practically  of  atmospheric  temperature,  we  see  that  the 
volume;  of  one  pound  has  been  reduced  from  approx- 


344       THE   TEMPERATURE-ENTROPY   DIAGRAM. 

imately  13  cu.ft.  to  approximately  2.6  cu.ft.  In  ot hoi- 
words,  the  volume  of  every  part  of  the  system  is  reduced 
to  about  one-fifth  of  that  which  would  be  needed  if  atmos- 
pheric air  had  been  used.  The  temperatures  ordinarily 
attain,  with  the  Allen  Dense  Air  Machine,  from  70°  to  90° 
F.  below  zero.  This  system  is  mostly  used  on  board  ship, 
and  the  air  is  used  to  a  certain  extent  in  a  regenerative 
manner,  i.e.,  the  coldest  air  just  leaving  the  motor 
passes  through  the,  ice-making  box,  and  then  somewhat 
warmer  next  enters  the  meat  chamber,  and  upon  leaving 
this  is  still  cold  enough  to  be  used  in  chilling  the  drink- 
ing water  butt.  Finally,  as  it  passes  on  its  way  to  the 
compressor,  it  is  used  to  chill  the  water-cooled  air  about 
to  enter  the  expander. 

The  amount  of  work  theoretically  is  the  same  in  the 
dense  air  as  in  the  natural  air  system,  but  because  of  the 
reduced  volume  the  compressor  and  motor  friction  losses 
may  be  smaller.  The  cost  is  less,  and  this,  combined 
with  the  decreased  floor  space,  makes  it  feasible  to  util- 
ize such  a  system.  Such  plants  are  still  used  in  pb'vs 
where  it  is  more  essential  to  guard  against  danger  aris- 
ing from  the  leakage  of  fluids,  such  as  ammonia,  than  to 
install  the  most  economical  plant,  as,  for  example,  on 
war  vessels. 

Ammonia  Refrigerating  Plant. — A  complete  cycle  of 
the  working  fluid  in  the  second  class  of  refrigerating 
plants,  where  a  saturated  vapor  is  used,  differs  mate- 
rially from  the  above,  as  the  substance  is  condensed 


DISCUSSION  OF  REFRIGERATING  PROCESSES.   345 

and  vaporized  during  the  process.  The  more  com- 
monly used  fluids  are  ammonia,  carbon  dioxide, 
and  sulphur  dioxide;  ammonia  being  used  most 
generally. 

Fig.  117  shows  diagrammatically  the  essential  features 
of  an  ammonia  refrigerating  plant.     It  consists  of  (1)  a 


FIG.  117. 

compressor  which  takes  the  low-pressure  vapor  from 
the  refrigerator  coils  and  delivers  it  at  some  higher 
pressure,  (2)  a  condenser  consisting  of  a  series  of  coils 
in  which  the  hot  gas  is  cooled  until  it  liquefies,  (3)  a 
storage-tank  containing  a  supply  of  ammonia  which 
remains  liquefied  under  the  high  pressure  at  atmos- 
pheric temperature,  (4)  an  expansion-valve  from  which 
the  liquid  emerges  under  reduced  pressure,  and  (5)  the 


346        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

refrigerator  coils  in  which  the  liquid,  under  reduced 
pressure,  is  vaporized  by  withdrawing  the  necessary 
heat  from  its  surroundings.  The  high  pressure  pre- 
vails from  the  delivery-valve  of  the  compressor  to  the 
expansion- valve,  and  low  pressure  from  the  lower  side 
of  the  reducing-valve  to  the  admission-valves  of  the 
compressor. 

The  refrigerator  coils  may  be  used  directly,  thus 
bringing  the  temperature  of  the  surroundings  down 
near  the  boiling  temperature  of  the  liquid,  or,  if  such 
a  low  temperature  is  not  desired,  the  coils  may  pass 
through  a  bath,  as  of  brine,  and  reduce  this  to  the 
desired  temperature.  The  cold  brine  is  then  circulated 
through  the  refrigerator.  This  latter  method  gives  a 
more  nearly  constant  temperature.  The  least  move- 
ment of  the  expansion-valve  causes  variations  in  the 
back  pressure,  and  hence  in  the  boiling  -temperature 
of  the  ammonia,  which  would  affect  the  surrounding 
air  if  used  directly,  but  which  would  be  absorbed  by 
the  large  heat  capacity  of  the  brine.  The  direct  sys- 
tem is,  however,  simpler  and  less  expensive  to  install 
and  to  maintain. 

As  long  as  any  liquid  remains  unvaporized  in  the 
refrigerator  coils  these  will  remain  at  the  temperature 
of  vaporization,  but  afterwards  the  pipes  assume  the 
higher  temperature  of  the  bath  or  surrounding  atmos- 
phere, and  then  begin  to  superheat  the  vapor  at  con- 
stant pressure.  This  superheating  is  still  further  in- 


DISCUSSION  OF  REFRIGERATING  PROCESSES.  347 

creased  in  the  pipe  leading  back  to  the  compressor. 
After  leaving  the  compressor  the  highly  superheated 
vapor  passes  to  the  condenser,  losing  part  of  its  super- 
heat at  constant  pressure  on  the  way.  This  process 
is  finished  in  the  condenser,  and  then  the  vapor  begins 
to  liquefy  at  the  temperature  corresponding  to  the  high 
pressure.  The  liquid  finally  emerges  cooled  to  the 
temperature  of  the  cooling  water  and  collects  in  the 
storage-tank  above  the  expansion-valve  ready  for  a 
new  cycle. 

The  corresponding  pv-  and  T^-cycles  are  shown  in 


FIG.  118. 

Fig.  118,  ab  represents  the  passage  of  the  liquid  through 
the  expansion-valve  along  a  constant-heat  curve,  dur- 
ing which  a  portion  of  the  liquid  r~  is  vaporized  and 
the  refrigerative  power  of  its  liquefaction  destroyed; 


348       THE   TEMPERATURE-ENTROPY   DIAGRAM. 

be  represents  the  vaporization  of  the  remainder  of  the 
liquid;  cd,  the  superheating  of  the  vapor  during  the 
last  part  of  the  refrigerator  coils  and  the  return  pipe 
to  the  compressor;  de'  represents  the  compression,  and 
e'h,  the  loss  of  superheat  by  conduction,  radiation,  etc., 
as  the  hot  gas  flows  along  the  pipe  to  the  condenser. 
If  the  compressor  were  two-stage,  with  intermediate 
cooling  down  to  atmospheric  temperature,  the  path 
followed  would  be  defgh.  The  liquefaction  is  repre- 
sented by  hi,  and  the  further  cooling  of  the  liquid  down 
to  atmospheric  temperature  by  w. 

To  decrease  the  work  required  to  compress  the  gas, 
attempts  are  made  in  various  types  of  compressors  to 
cool  it  during  compression  by  the  use  of  water-jackets 
or  by  the  direct  injection  into  the  cylinder  of  either 
liquid  ammonia  or  oil.  In  such  cases  the  compression 
is  no  longer  adiabatic,  but  of  the  form  pvn=plrln, 
where  n  will  depend  in  any  given  case  upon  the  amount 
of  heat  extracted  by  the  jackets  or  absorbed  by  the 
injected  fluid. 

The  temperature  of  the  entering  vapor  is  usually 
considerably  lower  than  that  of  the  cooling  water,  so 
that  heat  is  radiated  only  during  the  latter  part  of  the 
stroke,  when  the  temperature  of  the  vapor  greatly  ex- 
ceeds that  of  the  water.  The  compression  line  of 
indicator-cards  from  such  compressors  should  there- 
fore approximate  closely  the  adiabatic  curve  drawn 
through  the  commencement  of  the  compression  stroke 


DISCUSSION  OF   REFRIGERATING  PROCESSES.    349 

and  should  begin  to  fall  below  it  more  and  more  only 
as  the  discharge  pressure  is  approached. 

If  oil  of  the  same  temperature  as  the  entering  vapor 
is  injected  into  the  cylinder,  it  can  affect  the  temper- 
ature only  by  absorbing  heat  as  the  gas  is  compressed. 
The  specific  heat  of  the  oil  is  greater  than  that  of  the 
cylinder  walls,  and  possibly  conduction  occurs  some- 
what more  rapidly  from  vapor  to  oil  and  then  oil  to 
metal  than  it  would  directly  from  vapor  to  metal, 
especially  if  the  oil  is  in  a  finely  divided  state.  This 
can  only  result  in  changing  slightly  the  exponent  n  of 
the  compression  curve  pvn  =  plv1n. 

The  effect  of  injecting  liquid  ammonia  is  difficult  to 
describe  in  general  terms,  as  the  results  will  differ 
according  to  the  quantity  injected  and  the  various 
temperatures  of  the  liquid,  vapor,  and  cylinder  walls. 
If  the  cylinder  walls  are  assumed  to  be  non-conduct- 
ing and  the  injected  liquid  is  previously  cooled  to  the 
temperature  of  the  refrigerator,  the  superheated  vapor 
will  then  lose  its  superheat  and  in  so  doing  suffer  a 
drop  in  pressure,  as  the  cylinder  volume  is  momen- 
tarily constant.  At  the  same  time,  however,  the  in- 
jected liquid  will  be  partially  vaporized,  increased  in 
volume,  and  thus  effect  an  increase  in  pressure.  The 
resultant  effect  in  this  case  would  undoubtedly  be  a 
net  reduction  of  pressure  and  thus  decrease  the  work 
of  compression.  As,  however,  the  cylinder  walls  are 
good  conductors  when  in  contact  with  a  liquid,  enough 


350       THE  TEMPERATURE-ENTROPY   DIAGRAM. 

of  the  ammonia  might  thus  be  vaporized  to  produce  a 
net  increase  in  pressure.  If  the  liquid  is  injected  at 
the  same  temperature  as  the  entering  vapors,  it  is  at 
a  higher  temperature  than  that  of  saturated  vapor  at 
the  prevailing  pressure,  and  hence  will  partially  vapor- 
ize until  a  condition  of  equilibrium  is  established.  What 
the  final  conditions  of  pressure  and  temperature  will 
be  will  evidently  depend  upon  the  relative  weights  of 
liquid  and  vapor,  the  pressure,  and  the  temperatures 
of  vapor,  liquid,  and  cylinder.  In  either  of  the  above 
assumptions,  if  the  resultant  pressure  is  less  and  part 
of  the  liquid  still  remains  unevaporated,  the  work  of 
compression  would  be  still  further  decreased,  as  satu- 
rated vapors  transmit  heat  to  the  cylinder  walls  more 
rapidly  than  superheated  vapors. 

If  the  liquid  is  injected  into  the  suction-pipe  of  the 
compressor,  it  will  expand  at  the  prevailing  back 
pressure  and  reduce  the  temperature  down  to  that 
corresponding  to  that  pressure.  Whether  or  not  there 
results  a  net  diminution  in  volume  must  depend  upon 
the  amount  injected  and  the  quantity  of  heat  received 
from  external  sources.  Although  the  work  may  or 
may  not  be  decreased,  according  to  circumstances,  the 
temperature  will  at  least  be  decreased,  and  thus  the 
amount  of  necessary  cooling  water  diminished. 

It  is  theoretically  possible  to  effect  a  further  saving 
in  liquid  refrigerating  plants  by  changing  from  the 
throttling-curve  ab  (Fig.  118)  to  a  frictionless  adiabatic 


DISCUSSION  OF  REFRIGERATING  PROCESSES.    3ol 

expansion  ab' '•  that  is,  by  replacing  the  expansion- 
valve  with  an  auxiliary  cylinder  and  thus  utilize  the 
expansive  force  of  the  ammonia  to  help  run  the  com- 
pressor. The  refrigerative  power  of  the  ammonia 
would  be  increased  at  the  same  time,  since  the  amount 
vaporized  during  expansion  would  be  decreased  from 
kb  to  kb'.  It  is  possible  that  the  mechanical  com- 
plications thus  introduced  would  more  than  counter- 
balance the  thermodynamic  savings. 

It  has  also  been  suggested  that  the  loss  in  refriger- 
ative power  occasioned  by  the  expansion  ab  could  be 
diminished  by  reducing  the  temperature  of  the  liquid 
at  the  point  a.  Thus  the  gas  in  passing  from  the 
refrigerator  to  the  compressor  absorbs  from  the  atmos- 
phere the  heat  represented  by  the  area  under  Id.  If 
such  a  loss  is  unavoidable,  it  could  be  neutralized  by 
jacketing  the  return  pipe  with  the  liquid  ammonia 
about  to  be  fed  to  the  refrigerator,  thus  reducing  the 
temperature  of  the  latter  from  a  to  at  and  so  decreas- 
ing the  amount  vaporized  by  expansion  from  kb  to 
W. 

Conditions  for  Maximum  Efficiency. — The  ammonia, 
after  leaving  the  expansion  coil,  must  be  as  cold  as  the 
lowest  temperature  desired  in  the  refrigerating  plant, 
and  after  condensation  in  the  cooling  coils  cannot  have 
a  temperature  less  than  that  of  the  cooling  water.  Evi- 
dently the  minimum  temperature  range  is  covered  by 
these  two  factors,  so  that  the  minimum  amount  of  com- 


352       THE  TEMPERATURE-ENTROPY   DIAGRAM. 

prossion  work  will  be  obtained  when  the  ammonia  is 
cooled  just  to  the  minimum  temperature  and  is  com- 
pressed to  a  pressure  corresponding  to  the  temperature 
of  the  cooling  water.  Such  conditions  necessitate  the 
ammonia  entering  the  compressor  as  soon  as  it  is  com- 
pletely vaporized.  If  we  call  TI  the  upper  temperature 
and  T2  the  lower  temperature,  the  refrigcrative  effect 
per  pound  of  ammonia  will  be  represented  by  H2  —  q\ 
and  the  work  of  compression  will  be  represented  by 
H\  —  H<z,  so  that  the  maximum  efficiency  under  such 
conditions  will  be  represented  by  the  expression 


HI  —  /22 

From  this  expression  it  is  evident  that  the  efficiency 
increases  as  Hi  —  H2  diminishes.  This  can.be  accom- 
plished in  two  ways:  either  by  raising  the  lower  pres- 
sure or  diminishing  the  upper  pressure.  On  the  other 
hand,  a  decrease  in  the  lower  pressure  or  an  increase  in 
the  upper  pressure  will  cause  a  decrease  in  efficiency. 
It  is  evident  that  decreasing  the  back  pressure  makes 
it  possible  for  the  refrigerent  to  attain  the  temperature 
of  the  refrigerating  room  only  when  in  a  superheated 
condition.  This  gives  a  small  increase  in  the  refriger- 
ating effect  but  a  larger  increase  in  the  specific  volume 
of  the  ammonia,  so  that  less  ammonia  is  taken  into  the 
compressor  per  stroke  and  the  compressor  must  be 
speeded  up  to  produce  the  same  cooling  effect  as  before 


DISCUSSION   OF  REFRIGERATING  PROCESSES.  353 

the  change.  If,  on  the  other  hand,  the  lower  pressure 
is  left  unchanged  and  the  initial  pressure  increased,  the 
refrigerating  effect  per  pound  remains  constant,  and  the 
work  of  compression  is  increased  so  that  more  power 
would  be  required  to  drive  the  compressor. 

In  actual  practice  we  fail  to  realize  the  ideal  efficiency 
in  refrigerators  because,  (1)  all  of  the  heat  H2  —  qi  is  not 
absorbed  from  the  brine  or  the  refrigerating  room,  but 
partially  from  the  atmosphere,  so  that  (2)  the  temper- 
ature of  the  ammonia  entering  the  compressor  is  hotter 
than  that  of  the  returning  brine,  or  than  the  refrigerat- 
ing rooms,  thus  increasing  both  H2  and  H\,  and  there- 
fore increasing  the  difference  Hi  —  H2',  (3)  to  obtain 
fairly  rapid  flow  of  heat  from  the  ammonia  to  the  cooling 
water  the  discharge  pressure  must  exceed  the  theoretical 
minimum,  thus  increasing  the  work  of  compression. 

The  amount  of  refrigeration  of  any  given  temperature 
decrease  is  limited  only  by  the  speed  at  which  the  am- 
monia compressor  can  be  run.  The  efficiency  of  the 
operation  may,  however,  be  impaired  somewhat,  due  to 
the  greater  throttling  action  of  the  ports  and  valves. 

To  obtain  refrigeration  of  greater  intensity  the  throttle 
valve  must  be  throttled  still  more.  This  will  result  in  a 
reduction  of  efficiency,  since  not  only  is  the  refrigeration 
decreased  but  the  work  of  compression  is  increased. 

Absorption  Refrigeration.— There  are  two  ways  in 
which  ammonia  may  be  returned  from  the  lower  to  the 
higher  pressure  level.  One  method  we  have  already 


354       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

discussed ;  namely,  that  of  compressing  it  in  a  suitable 
compressor  and  then  condensing  by  means  of  cold  water. 
The  other  method  consists  in  using  an  absorbent,  as,  for 
example,  water,  which  is  used  in  all  ordinary  operations 
and  which  possesses  the  power  of  absorbing  many  times 
its  own  volume  of  ammonia  gas,  and  then  using  a  small 
pump  to  force  the  concentrated  solution  from  the  lower 
pressure  to  the  upper  pressure  and  deliver  it  to  a  boiler 
or  generator,  where  the  ammonia  is  liberated  from  the 
water  by  the  application  of  heat.  According  to  the 
temperature  at  which  the  absorption  takes  place,  a 
pound  of  water  will  absorb  from  a  fraction  of  a  pound 
to  something  more  than  two  pounds  of  ammonia.  This, 
expressed  in  gaseous  volume,  shows  a  decrease  of  two 
or  three  hundred  volumes  to  one,  i.e.,  the  action  of 
water  is  equivalent  to  a  perpetual  vacuum.  In  the 
generator  the  application  of  heat  evaporates  not  only 
the  ammonia  but  also  a  considerable  amount  of  water 
vapor,  which  must  necessarily  be  eliminated  before  the 
ammonia  can  be  used  as  a  refrigerating  agent.  On  the 
top  of  the  generator  is  a  tower-like  structure  consisting 
of  a  series  of  troughs  and  passage-ways  so  designed  as 
to  return  any  condensate  to  the  boiler  without  hindering 
the  upward  passage  of  the  vapors.  By  the  radiation  of 
heat  and  by  the  inrush  of  the  cold  concentrated  ammo- 
nia solution  from  the  absorber,  the  ascending  column  of 
vapors  is  chilled  and  the  less  volatile  steam  suffers  the 
greater  condensation.  At  the  same  time  the  incoming 


DISCUSSION  OF  REFRIGERATING  PROCESSES.    355 

solution  is  heated  and  a  portion  of  the  more  volatile 
ammonia  is  driven  off  and  passes  out  at  the  top  of  the 
tower,  together  with  the  ammonia  which  has  come  from 
the  generator.  Even  now  the  gas  is  mixed  with  a  cer- 
tain small  quantity  of  steam  which  must  be  eliminated. 
This  is  accomplished  in  a  cooling  coil  known  as  the  "rec- 
tifier" and  so  drained  that  any  condensate  flows  back 
into  the  tower  or  analyzer.  This  rectifier  is  usually 
placed  at  some  point  exposed  to  the  atmosphere,  such  as 
the  roof  of  a  building,  and  is  cooled  by  a  small  quantity 
of  water  trickling  down  over  the  pipes,  the  object  being 
to  utilize  the  heat  of  vaporization  of  the  water,  which 
vaporization  is  furthered  by  any  air  currents  which  may 
strike  against  the  coils.  From  this  point  on  the  am- 
monia is  practically  free  from  water,  so  that  the  next 
portion  of  the  cooling  coil  drains  forward  into  the  am- 
monia tank  in  which  the  liquid  ammonia  is  stored  at 
practically  atmospheric  temperature  until  it  passes 
clown  through  the  expansion  valve.  This  portion  of 
the  apparatus  is  identical  in  both  the  compressor  and 
absorption  systems,  and  makes  use  of  the  cooling  effect 
occasioned  by  the  unresisted  expansion  from  high  to 
low  pressure.  This  operation  produces  a  slight  vapor- 
ization of  the  ammonia,  and  to  that  extent  destroys  its 
refrigerative  value.  The  remaining  liquid  passes  into 
the  refrigerating  coils  which  may  be  located  either 
directly  in  the  refrigerating  rooms,  or  inserted  in  a  brine 
solution  in  case  indirect  refrigeration  is  desired.  After 


356        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

being  vaporized,  the  ammonia  passes  into  the  absorbent, 
as  mentioned  above.  During  absorption  the  ammonia 
liberates  about  900  B.T.U.  per  pound.  The  resulting 
increased  temperature  of  the  mixture  from  such  a  gen- 
eration of  heat  would  prevent  the  absorption  of  ammo- 
nia, so  that  this  heat  of  absorption  must  be  eliminated 
from  the  generator  by  the  circulation  of  cold  water  from 
suitable  pipe  coils.  It  ic  evident  that  the  water  which 
is  left  behind  in  the  generator,  when  the  ammonia  is 
driven  off  by  heat,  must  be  drawn  off,  otherwise  the 
generator  would  soon  be  filled,  and  similarly  the  water 
which  is  taken  out  of  the  absorber  must  be  replaced,  or 
else  the  absorber  will  soon  empty.  Both  of  these  needs 
can  be  fulfilled  by  permitting  the  water  to  return  from 
the  generator  to  the  absorber.  Naturally  the  weakest 
solution  will  be  found  at  the  bottom  of  the  generator, 
so  that  the  return  pipe  takes  the  water  from  that  point. 
We  have  now  two  streams  passing  in  opposite  directions; 
a  rich  solution  passing  from  the  analyzer  to  the  genera- 
tor, which  must  be  heated  in  order  to  give  off  the  am- 
monia, and  a  weak  solution  passing  from  the  generator 
to  the  absorber  which  must  be  cooled  in  order  to  be  able 
to  absorb  the  ammonia.  Evidently,  then,  both  of  these 
purposes  can  be  effected  by  permitting  the  two  streams 
to  flow  in  opposite  directions,  but  in  thermal  contact, 
so  that  the  rich  solution  will  be  warmed  by  taking  up 
the  heat  of  the  weak  solution.  Of  course  this  necessi- 
tates that  a  force  pump  be  located  between  the  absorber 


DISCUSSION 'OF  REFRIGERATING  PROCESSES.    357 

and  the  point  at  which  this  heat  is  received  by  the  strong 
solution.  This  device,  which  transfers  the  heat  from 
the  weak  to  the  strong  solution,  is  known  as  an  "inter- 
changer"  or  "regenerator."  Such  a  system  appears 
complicated  at  first  sight,  but  after  a  little  thought  be- 
comes fairly  simple  if  we  simply  stop  to  realize  that  the 
compre&sor  is  being  replaced  by  a  water  conveyor,  which 
must  first  be  cooled  and  then  heated  in  order  to  ab- 
sorb and  deliArer  the  ammonia,  and  that  most  of  the 
complexities  of  the  mechanism  arise  in  carrying  out 
these  two  functions  of  heating  and  cooling,  and  the  fur- 
ther function  of  separating  the  water  vapor  entirely 
from  the  ammonia. 

There  are  really  three  separate  cycles  being  followed 
throughout  the  mechanism.  There  is  the  cycle  of  the 
water,  the  cycle  of  the  ammonia,  when  free  and  when 
absorbed,  and  finally,  in  case  of  indirect  refrigeration, 
there  is  also  a  brine  solution.  These  different  cycles 
or  paths  may  be  outlined  as  follows:  The  weak  solu- 
tion starts  at  the  generator,  passes  through  the  exchanger 
to  the  absorber  where  it  meets  the  gaseous  ammonia,  and 
the  two  unite.  The  strong  solution  is  then  taken  by  the 
pump,  forced  through  the  exchanger,  this  time  on  the 
high  pressure  side  of  the  pump,  and  separated  from  the 
hot  weak  solution  only  by  the  thickness  of  the  piping, 
and  then  enters  the  top  of  the  analyzer.  At  this  point 
most  of  it  falls  downward  toward  the  generator,  a  small 
portion  is  perhaps  vaporized  and  passes  up  into  the  rec- 


358       THE  TEMPERATURE-EXTROPY  DIAGRAM. 

tifier,  and  then  falls  back  again  into  the  generator.  ,-o 
that  with  the  exception  of  a  few  particles  which  keep 
whirling  around  in  eddy  currents  between  the  analyzer 
and  the  rectifier,  most  of  it  passes  down  into  the  gener- 
ator, and,  having  left  all  the  ammonia  behind,  is  drawn 
off  as  a  weak  solution  and  begins  its  cycle  once  more. 
The  ammonia  leaves  the  generator  in  conjunction  with 
the  steam  vapor,  passes  up  through  the  analyzer  under- 
going partial  condensation  and  re-evaporation,  but  in 
the  main  passing  onward  to  the  rectifier,  where  it  again 
undergoes  partial  condensation  and  escapes  from  the 
rectifier  entirely  free  from  water.  It  then  goes  through 
the  condenser,  throttle  valve,  expansion  coils,  into  the 
absorber,  where  it  unites  with  the  weak  solution  and 
travels  the  path  already  described,  namely,  from  the 
absorber  through  the  pump  and  exchanger  to  the  ana- 
lyzer. In  case  of  brine  refrigeration,  the  ammonia  ex- 
pansion coils  are  placed  in  the  brine  tanks.  Then  the 
cool  brine  is  forced  through  suitable  piping  to  the  refrig- 
erating rooms  where  it  picks  up  heat  and  returns  once 
more  to  the  cooling  tanks. 


CHAPTER  XVIIJ. 
DISCUSSION   OF  KELVIN'S  WARMING  ENGINE. 

The  Warming-engine. — The  third  possibility  of  the 
reversed  cycle,  viz.,  the  utilization  of  the  heat  deliv- 
ered at  the  upper  temperature  for  heating,  was  pointed 
out  by  Lord  Kelvin;  the  idea  being  that,  given,  say, 
a  definite  quantity  of  steam  for  heating  purposes,  a 
greater  heating  effect  could  be  obtained  by  utilizing  the 
steam  to  run  an  engine  and  compressor  system  and 
then  diverting  the  exhaust  steam  of  the  engine  and  the 
heated  fluid  of  the  compressor  to  heating  purposes,  than 
by  a  direct  application  of  the  steam  itself. 

The  explanation  of  this  fact  is  that  in  the  one  case 
the  availability  of  the  heat  to  perform  work  at  the  high 
temperature  is  utilized,  while  in  the  other  it  is  lost. 
When  heat  is  transferred  by  conduction,  radiation,  etc., 
from  a  hot  body  to  a  colder  body,  the  entropy  of  the 
hot  body  decreases  and  that  of  the  cold  body  increases. 

But  when  an  adiabatic  transfer  of  energy  occurs  by 
changing  heat  first  into  work  and  then  back  into  heat, 
by  alternate  expansion  and  compression,  the  entropy  of 
each  part  of  the  system  remains  unchanged  and  the 
temperature  of  the  hot  body  is  decreased  while  that  of 

the  cold  body  is  raised.    The  action  of  such  a  system 

359 


360        THE    TEMPERATURE-ENTROPY  DIAGRAM. 


can  best  be  illustrated  by  assuming  an  ideal  system  from 
which  all  friction  and  conduction  losses  are  eliminated, 
and  comparing  the  relative  heating  effects  obtained  by 
direct  heating  and  by  using  the  warming  engine. 

Problem. — A  pcund  of  dry  steam  at  150  pounds  abso- 
lute pressure,  if  used  in  a  radiator  and  trapped  out  at 
room  temperature,  say  70°  F.,  would  yield  1155  B.T.U. 
Assuming  this  steam  to  operate  upon  the  Rankine  cycle, 


FIG.  119. 

discharging  non-condensing  into  the  heating  system  at 
17  Ibs.  absolute,  the  exhaust,  before  leaving  the  radia- 
tor as  water  at  70°  F.,  would  yield  1095.3  B.T.U.,  while 
59.7  B.T.U.  would  be  developed  into  work. 

Let  the  steam-engine  furnish  the  work  required  to  drive 
a  compressor-motor  unit  which  operates  with  one  pound 
of  air.  The  compressor  receives  the  air  at  atmospheric 
temperature  and  discharges  it  at  some  higher  temper- 
ature T\  into  radiator  coils,  where  it  is  cooled  under 


DISCUSSION  OF   KELVIN'S  WARMING  ENGINE.    361 

constant  pressure  p\  to  room  temperature  TR  (Fig.  119). 
It  next  passes  to  the  motor  where,  after  expanding  to  a 

T  •  T 
final    temperature    Tr  =••    "      R  ,  it    exhausts    into    the 

atmosphere  and  a  fresh  charge  of  warmer  air  is  taken 
by  the  compressor.  The  effect  is  equivalent  to  taking 
from  the  atmosphere  per  pound  of  air  an  amount  of 


heat  equal  to  Cp(Ta-Tx\  or  0.2375^-r      B.T.U. 

1  1 

The  heat  rejected  to  the  radiator  equals  0.2375(71  -  TR} 
B.T.U.,  while  the  work  required  to  drive  the  unit  is 


equal  to  0.2375(7i  -TR)\l-jr)-     Therefore,  the  ratio 

-I  rp 

of  heat  delivered  to  work  input  is  — — ^-,  or  ~ — ~. 

i      __2  '  —     a 

From  this  it  follows  that  the  air  effects  a  heating  equal 
to  -fp ni~  times  the  heat  developed  into  work.     But 


or 


JH 

T        ATJ 


If  we  assume  the  temperature  of  the  outside  air  to  be 
0°  F.  a  substitution  of  the  values  Ta  -459.5,  TR  =517.  -I. 
Cp  =  0.2375  and  J//-59.7,  gives  T  t=  996.0°  F.  abs.  or 
t}  -536.5.  From  this  it  follows  that  the  air  rejects 
Cp(T1-77flH0.2375(530.5-70)  =  110.8  B.T.U.  in  pass- 


362       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

ing  through  the  radiator.  This  represents  a  gain  of 
110.8-59.7  or  51.1  B.T.U.  over  the  energy  required  to 
drive  the  compressor-motor  unit,  and  is  therefore  the 
heat  gained  from  the  low  temperature  supply  of  the 
atmosphere. 

In  the  above  problem  an  unnecessary  limitation  was 
imposed  by  assuming  one  pound  of  air  per  pound  of 
steam.  If  n  pounds  of  air  are  compressed  per  pound 

of  steam  each  pound  will  receive  but  -  as  much  work, 

so  that  the  delivery  temperature  will  be  something  less 
than  T\,  say  7Y,  which  may  be  obtained  from  the 
equation 

T  '     T      T     ,Ta'TR      ^H 

J-  1    —*a—lR  H -ffTi—  -~Cr' 

1  1  71  •  U  p 

Assuming  n  =  10,  the  delivery  temperature  reduces  to 
624.0°  F.  abs.,  or  164.5°  F.  The  heat  rejected  by  the 
air  in  the  radiator  now  equals  10x0.2375  (164.5-70)  = 
224.4  B.T.U.  This  represents  a  gain  of  224.4-59.7  = 
164.7  B.T.U.,  which  is  the  heat  obtained  from  the  low 
temperature  supply  of  the  external  atmosphere. 

It  should  be  noticed  that  although  by  increasing  the 
amount  of  air  circulated  the  quantity  of  low  temperature 
heat  which  becomes  available  has  been  raised  from 
51.1  B.T.U.  to  164.7  B.T.U,  this  has  been  obtained  at 
the  expense  of  a  larger  compressor,  because  less  heating 
is  done  per  unit  weight  of  air  as  the  upper  temperature 
TI  gradually  diminishes  in  magnitude  with  increasing 


DISCUSSION  OF  KELVIN'S   WARMING    ENGINE.   363 

weight  of  air.  Evidently  the  limiting  condition  is  reached 
when  TI  becomes  equal  to  TR,  when  there  results  the 
maximum  heating  effect  from  the  air,  viz., 


R  Q,  R  Q, 

In  the  case  of  the  above  problem  this  amounts  to 
^pX  59.7  --=  65.64  X  59.7=--  391.9  B.T.U.  The  com- 
bined heating  effect  from  the  steam  and  air  is  thus 
1487  B.T.U.  in  place  of  the  1155  B.T.U.  from  the  steam 
alone. 

An  investigation  of  the  general  expression  for  the 
heat  gained  from  external  sources  per  pound  of  air, 

71          AJ-t 

j,  _afj, ,  indicates  two  methods  of  increasing  this 

result.  This  first,  which  has  already  been  mentioned, 
consists  in  making  T\  approach  TR  in  value  by  increas- 
ing the  quantity  of  air  circulated.  The  second  consists 
in  making  Ta  approach  TR.  This  is  ordinarily  be- 
yond the  control  of  the  engineer,  who  must  take  the 
temperature  of  the  external  air  as  he  finds  it,  but  it  at 
least  teaches  that  the  smaller  the  increase  in  tempera- 
ture through  which  the  external  air  must  be  raised  the 
greater  the  gain  from  the  warming  engine. 

An  inspection  of  Fig.  119  makes  this  evident.  The 
lower  the  temperature  T\  the  higher  the  temperature  Tx, 
since  T\  •  T x  =  Ta-TR,  and  therefore  the  smaller  the  area 


364        THE    TEMPERATURE-ENTROPY  DIAGRAM. 
AW  relatively  to  the  area  Q->.     Thus  the  ratio  of  areas 
increases  from  ~ — ^-  to  ™ — ^~-  as  the  upper  tem- 
perature decreases  from  T\  to  TR.     Further,  as  Taap- 

Ta 
proaches  TR  the  value  of  ™ — -y-  increases  indefinitely. 

General  Discussion. — We  are  now  in  a  position  to 
give  a  more  general  discussion  of  this  subject  in  thai  we 
avoid  the  special  characteristics  of  any  particular  sub- 
stance by  assuming  the  working  substances  in  both  the 
engine  and  the  compressor-motor  unit  to  operate  upon 
the  direct  and  reversed  Carnot  cycles  respectively. 

Let  T\  be  the  temperature  of  a  limited  supply  of  heat, 
Qi,  and  T2  that  of  an  unlimited  supply,  say,  of  the  at- 
mosphere, and  T3  some  intermediate  temperature  to 
which  a  room  is  to  be  warmed.  A  Carnot  engine  work- 

m    rp 

ing  between  T\  and  T%  would  perform  the  work  Qi — ^ — 1 

rp 

and  reject  the  heat  Qi^f.     Suppose.,  this  work  to  be 

expended  upon  an  air-cornprcssor,  heating  the  air  from 
T2  to  TV  As  the  entropy  of  the  air  does  not  increase 
during  adiabatic  compression,  the  entropy  of  the  heat 
to  be  taken  from  the  air  so  compressed  must  be  equal 
to  the  heat  equivalent  of  the  work  of  compression 
divided  by  the  increase  in  temperature,  or 

j  3 ~ T2      T\  7*3  —  T2 ' 


DISCUSSION  OF  KELVIN'S   WARMING  ENGINE.    365 

hence  the  supply  of  air  taken  into  the  compressor  must 
have  available  the  heat 

,         .^2  Ti  —T3 


Th?  boat  available  in  tli3  hot  air  at  temperature 
will  then  be 

fri      FTI    _  fji  rji    _  rji 


which  could  also  be  obtained  by  multiplying  the  de- 
creased entropy  of  the  air  by  the  final  temperature  T3. 

Thus 

T      T  —T 

r\      _    A   i        m    _s\   x  3      J   1         J  3 
WA  —  J(Pz'  1  3~*viT   '  T        T    ' 

1  1      L  3  ~~  '•*•  2 

The  total  quantity  of  heat  delivered  to  the  room  thus 
becomes  the  sum  of  that  rejected  by  both  engine  and 
compressor,  or 

rp  fri  /TT      rn    _  /7i 


The  increase  in  heating  power  over  that  obtained  by 
the  use  of  the  steam  alone  is  thus 


366       THE    TEMPERATURE-ENTROPY  DIAGRAM. 

or  equal  to  the  heat  obtained  from  the  low  temperature 
air.  An  examination  of  this  last  formula-  shows  that  if 
the  upper  and  lower  temperatures,  7\  and  T2,  are  fixed, 
the  gain  will  be  greater  the  smaller  the  value  of  T3,  so 
that  when  the  range  T3  —  T-2  is  small  the  gain  may  be 
many  times  the  original  quantity  of  heat.  The  gain 
Qs —Qi  represents  the  extreme  difference  obtained  by 
supposing  in  one  case  all  the  availability  to  be  utilized 
and  in  the  other  that  none  of  it  is  utilized,  or  that  the 
hot  body  simply  expanded  along  a  constant  heat-curve 
until  T3  is  reached,  thus  suffering  an  increase  of  its 
own  entropy.  In  practice  the  actual  difference  would 
be  diminished  from  both  sides,  the  maximum  value  of 
Q3  being  impossible  to  attain,  due  to  radiation,  con- 
duction, and  friction  losses,  and  the  minimum  value  Qi 
would  always  be  exceeded,  as  the  heat  contained  in  the 
air  would  be  partially  utilized. 

This  maximum  gain,  Qs—  Qi,  can  be  illustrated  by 
means  of  the  T<£-diagram,  as  shown  in  Fig.  120.  Let  ah 
represent  the  quantity  of  heat  Qt  at  the  temperature 
TI.  T2  is  the  temperature  of  the  atmosphere  and  T3 
that  in  the  room.  A  Carnot  engine  working  between 
7\  and  T3  would  perform  the  work  ac  and  exhaust  the 
heat  db,  the  temperature  of  the  exhaust  having  been 
lowered  from  Tl  to  T3,  but  the  entropy  remaining  con- 
stant at  a'b.  If  no  work  is  performed,  but  the  hot  body 
permitted  to  expand  along  the  constant  heat-curve  H, 
the  heating  effect  at  T3  will  be  de  =  ab  =  Ql.  Let  us  rep- 


DISCUSSION  OF  KELVIN'S  WARMING  ENGINE.   367 

resent  the  change  in  the  condition  of  the  air  at  the  left 
of  the  line  aa':  Let  mn(  =  ac)  represent  the  work  of 
the  compressor  in  heat-units.  As  this  has  to  bridge 


o' 


FIG.   120. 

over  the   temperature   interval   T3  —  T2,   its   width  or 
ac 


entropy  will  be  fa' 


The  heat  originally  con- 


tained  in  the  air  thus  compressed  is  therefore  shown 
by  nf.  The  total  heating  effect  is  thus  ma'  +db,  which 
is  greater  than  de  by  the  area  nf. 


368       THE  TEMPERATURE-ENTROPY  DIAGRAM. 

To  determine  the  position  of  the  point  m,  it  is  con- 
venient to  construct  the  rectangular  hyperbola  as  pass- 
ing through  a,  and  so  proportioned  that  when  inter- 
sected by  isothermals  T3,  T3',  etc.,  the  rectangles  mn, 
m'n,  etc..  thus  determined  will  be  equal  to  the  work 
performed  by  the  Carnot  engine,  ac,  ac',  etc.,  respec- 
tively. It  should  be  noticed  that  as  T3  approaches  T2, 
the  quantity  of  heat  nf,  nf,  etc.,  utilized  from  the  atmos- 
phere increases  indefinitely. 

A  Gas  as  the  Working  Fluid. — The  above  general 
discussion  is  valuable  as  denning  a  maximum  standard 
of  reference,  but  the  Carnot  cycle  is  not  feasible  in 
practice,  and  must  be  replaced  by  the  Joule  cycle  for 
gases  and  by  the  Rankirie  cycle  for  saturated  and  su- 
perheated vapors.  An  illustration  of  the  use  of  the 
Joule  cycle  with  air  has  already  been  given  in  the  above 
problem,  which  shows  that  the  theoretical  gains  with 
air  heating  are  probably  not  sufficient  to  offset  the  cost, 
bulk,  maintenance  and  inconvenience  of  a  compressor- 
motor  unit  installed  for  this  purpose  alone.  That  is, 
the  heating  problem  is  different  from  that  of  refriger- 
ation. There  is  but  one  way  to  obtain  the  necessary 
refrigeration  and  the  expense  must  be  met,  but  in  heat- 
ing the  direct  method  is  the  simplest  and  most  obvious, 
and  the  warming  engine  must  justify  itself  by  showing 
a  considerable  thermal  saving.  Of  course  there  alwavs 
exists  the  interesting  possibilities  of  obtaining  the  heat- 
ing as  a  by-product  of  the  refrigerating  system,  or  of 


DISCUSSION   OF  KELVIN'S  WARMING  ENGINE.   369 

using  the  same  plant  for  cooling  in  summer  and  heating 
in  winter.  The  same  advantages  exist  here  in  the  use 
of  a  dense  air  system  as  in  the  case  of  refrigeration. 

A  Saturated  Vapor  as  the  Working  Fluid.— The  same 
qualities  which  make  the  use  of  a  saturated  vapor,  as 
ammonia,  advantageous  for  refrigeration,  hold  good  in 
the  case  of  the  warming  engine,  viz.,  the  large  heat 
capacity  and  the  small  volume. 

Here  the  cooling  coils  of  the  refrigerating  system 
would  be  replaced  by  large  radiating  surfaces  properly 
distributed  about  the  building  to  be  heated,  and  if  no 
refrigeration  was  desired  the  ammonia  heating  coils 
(evaporation  coils)  could  be  exposed  to  the  atmosphere, 
or  better  still,  if  convenient,  be  immersed  in  some 
nearby  river  or  ocean  and  thus  draw  heat  from  this 
limitless  supply.  Due  to  the  high  vapor  tension  of 
the  ammonia  the  temperatuie  in  the  radiator  coils 
would  be  less  than  in  steam  coils,  so  that  the  rate  of 
heat  flow  would  be  less  than  in  steam  heating  and 
consequently  a  greater  area  of  radiating  surface  would 
be  required. 

The  wanning  unit  would  consist  of  a  steam  engine, 
running  non-condensing  and  exhausting  into  steam 
heating  coils,  which  operates  a  direct-connected  am- 
monia (or  other  saturated  vapor)  compressor.  The 
heating  would  be  effected  by  the  exhaust  steam  at 
about  220°  F.  and  by  the  high-pressure  ammonia  at 
about  100°  F.  The  low  temperature  heat  would  be 


370        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


obtained   from  water  probably  at  a  temperature  of 
about  40°  F. 

Problem. — Assuming  that  the  ammonia  follows  the 
reversed  Rankine  cycle  between  the  pressure  limits 
thus  defined,  with  the  exception  that  cooling  along  the 
liquid  line  is  replaced  by  the  throttling  curve,  the  low 
temperature  heat  picked  up  at  40°  F.  would  equal 
ioo  =9  +  534-75-468  B.T.U.  The  work 


100 


15=210.70 


P=  72.59 


100 1 459.5 


FIG.  121. 


required  to  drive  the  compressor  (see  Fig.  121)  would 
be  Hi  —  H2.    To  obtain  H\  we  have  <£i  =  </>2,  or 


486 


whence 


i  =  643.9°  F.  abs.  =  184.4°  F. 


DISCUSSION  OF  KELVltfS  WARMING  ENGINE.  371 
Therefore 

#i  =561  +0.54x84.4-606.6  B.T.U., 
and 

Hi  -  II 2  =  606.6  -  543  =  63.6  B.T.U., 

so  that  the  net  heat  gained  from  the  operation  would 
be 

468-63.6  =  394.4  B.T.U.  per  pound  of  NH3. 

Electric  Heating. — Another  possibility  of  the  warm- 
ing engine  is  the  economical  utilization  of  electrical 
power  for  heating  purposes.  The  direct  utilization  of 
electric  power  in  resistance  heaters  is  much  less  econom- 
ical than  the  customary  furnaces,  hot  water  and  steam 
heaters,  but  if  the  electric  power  is  used  to  operate 
an  electric  motor-compressor  unit  it  is  possible  that 
the  efficiency  of  the  warming  engine  may  bring 
such  a  usage  within  the  realm  of  economic  pos- 
sibilities. 

Thus,  imagine  the  compressor  of  the  preceding  prob- 
lem driven  by  such  a  motor,  then  assuming  no  losses, 
the  heating  accomplished  by  the  warming  engine  would 

468 

be  :nr-;:  =  7.36  times  that  of  the  heat  equivalent  of  the 
63.6 

electrical  energy.  Of  course  this  full  theoretical  value 
would  riot  be  realized  in  practice  due  to  stray  power 
and  friction  losses,  but  it  should  be  noted  that  such 
losses  are  not  a  direct  thermal  loss  as  they  possess  the 


372        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

same  heating  effect  whether  obtained  in  the  motor 
windings  and  the  bearing  surfaces  or  in  a  resistance 
coil,  but  that  the  only  "loss"  which  they  signify  is 
a  limitation  of  the  amount  of  increase  to  be  obtained 
from  the  warming  engine.  It  is  possible  that  the  net 
returns  would  be  sufficient  to  bring  the  cost  of  elec- 
tricity within  competitive  range  with  coal,  especially 
as  the  greater  demand  for  such  heating  current  would 
come  during  the  day  so  that  the  manufacturer  could 
afford  to  sell  the  current  at  low  rates  in  order  to  in- 
crease the  load  factor  of  the  central  station,  and  thus 
lower  the  cost  of  power  to  all  consumers  and  in  this 
way  help  to  extend  the  applications  of  electncity  in 
other  directions. 

"Thermodynamically  our  present  methods  of  heating 
are  most  wasteful  and  subject  to  great  improvements, 
and  sooner  or  later  as  the  limitations  of  our  natural 
resources  begin  to  be  felt  a  more  economical  method 
will  be  developed.  When,  therefore/'  to  quote  the 
words  of  Prof.  Cotterill,  "we  warm  our  houses  by  the 
direct  action  of  heat  from  combustible  bodies,  we  waste 
by  far  the  greater  part  of  it  by  making  no  use  of  the 
high  temperature  at  which  the  heat  is  generated,  a 
small  quantity  of  heat  at  high  temperature  being  ideally 
capable  of  raising  a  large  quantity  to  a  moderate  tem- 
perature." 

"It  is  interesting,  and  may  some  day  be  useful," 
says  Prof.  Ewing,  "to  recognize  that  even  the  most 


DISCUSSION  OF  KELVIN'S  WARMING   ENGINE.     373 

economical  of  the  usual  methods  employed  to  heat 
buildings,  with  all  their  advantages  in  respect  of 
simplicity  and  absence  of  mechanism,  are  in  the 
thermodynamic  sense  spendthrift  modes  of  treating 
fuel." 


CHAPTER  XIX. 

ENTROPY  ANALYSIS   IN  THE   BOILER  ROOM. 

The  Steam-boiler. — The  boiler  is,  thermally,  the  least 
economical  member  in  any  steam  plant  installed  for  the 
production  of  power.  The  hot  gases  passing  over  the 
heating  surfaces  must  be  of  considerably  higher  tem- 
perature than  the  water  in  the  boiler,  so  as  to  insure 
a  rapidity  of  heat  transference  commensurate  with 
reasonable  development  of  power.  The  higher  there- 
fore the  boiler  pressure,  other  things  being  equal,  the 
greater  the  amount  of  heat  carried  off  in  the  exhaust 
gases.  Practice  shows  that  the  heat  lost  through  the 
setting  and  up  the  stack  varies  roughly  from  twenty  per 
cent  to  fifty  per  cent  of  the  heat  of  combustion  of  the 
fuel.  The  mere  statement,  however,  that  the  boiler 
efficiency  varies  from  sixty  per  cent  to  eighty  per  cent 
does  not  present  the  total  wastefulness  of  the  boiler 
when  the  steam  is  to  be  used  in  the  production  of  power 
and  not  for  mere  heating  and  industrial  purposes. 

Suppose, for  example,  that  two  boilers  develop  thermal 
efficiencies  of  seventy  per  cent  and  sixty-five  per  cent 
while  supplying  steam  at  100  Ibs.  absolute  and  250  Ibs.  ab- 
solute, respectively,  to  two  condensing  engines  exhaust- 
ing at  1  Ib.  absolute  back  pressure.  The  second  boiler 

374 


ENTROPY  ANALYSIS  IN  THE  BOILER  ROOM.     375 

is  really  the  more  efficient  thermally.  The  exhaust 
pressure  is  determined  by  the  effectiveness  of  the  corn- 
denser  and  the  vacuum  pumps  and  is  beyond  the  con- 
trol of  the  boiler,  and  is  therefore  made  equal  in  this 
illustration,  but  the  pressure  during  admission  is  deter- 
mined by  the  boiler  and  the  effects  of  its  variation  must 
be  charged  to  the  boiler  alone.  The  temperature  corre- 
sponding to  the  boiler  pressure  determines  the  motivity 
or  availability  of  the  heat  absorbed  in  the  boiler.  The 
temperatures  corresponding  to  250,  100  and  1  pounds 
are  401.1,  327.9  and  101.8  degrees  Fahrenheit,  so  that 
the  motivities  in  the  two  cases  cited  above  become 


327.9-101.8  401.1-101.8 

327.9+459.5  401.1+459.5' 


or  28.7  per  cent  and  34.8  per  cent.  So  that  the  net 
motivity  of  the  heat  of  combustion  of  the  coal  is  .70  X 
28.7  and  .65X34.8,  or  20.1  and  22.6  respectively.  It 
is  thus  evident  that  the  boiler  possessing  the  greater 
"boiler  efficiency"  may  really  be  the  less  efficient  when 
its  effect  upon  the  availability  of  the  heat  is  considered. 
Loss  of  Availability  between  the  Fire  and  the  Steam.— 
Besides  the  direct  heat  losses  of  30  and  35  per  cent 
quoted  in  the  above  problem  there  is  a  further  loss  due 
to  drop  of  temperature  of  49.9  and  42.4  per  cent  respec- 
tively. That  is,  the  loss  in  efficiency  due  to  drop  in 
temperature  between  the  hot  gases  in  the  furnace  and 


376       THE   TEMPERATlfRE-ENTROPY  DIAGRAM. 


the  steam  in  the  boiler  is  roughly  one  and  a  half  times 
that  due  to  the  direct  heat  losses. 

The  loss  of  availability  of  the  heat  in  a  boiler  is 
clearly  illustrated  by  the  temperature  entropy  diagram, 
,as  shown  in  Fig.  122.  Let  abdc  represent  a  certain 
quantity  of  heat  Q  which  leaves  the  gases  of  the  fur- 
nace at  the  temperature  TF,  and  let  a  We  represent 


Loss^AvailaMli,, 


FIG.  122. 

this  same  quantity  of  heat  after   it  has  entered  the 
water  cf  the  boiler  at  the    temperature  Ts.     As  the 

T 
temperature  has  decreased  in  the  ratio  -^-  the  entropy 

T 
factor  must  have  increased  in  the  ratio  ™-,  so  that  if 

1  a 

the  original  entropy  cd  possesses  the  magnitude  <£,  the 

T 
final  entropy  cd'  must  possess  the  entropy  jf-  •  </>.    Thus 


ENTROPY  ANALYSIS  IN  THE  BOILER  ROOM.  377 
while  the  entropy  of  the  hot  gases  in  the  furnace  has 
decreased  by  the  amount  ^>=-m-t  the  entropy  of  the 

m  r\ 

steam  has  increased  by  the  amount   ^f-  •  $  =  ^- ,  and 

J-  8  1   8 

there  has  therefore  resulted  a  net  increase  in  the  entropy 
of  the  system  of  the  amount  </>(  -™ —  1 J . 

If  now  Tc  represent  the  temperature  of  the  condenser, 
i.e.,  the  coldest  temperature  practically  attainable,  the 
area  efdc  represents  the  heat  theoretically  unavailable 
for  work,  while  the  heat  Q  is  still  contained  in  the  gases, 
while  efd'c  represents  the  heat  no  longer  available  after 
the  heat  Q  has  entered  the  steam.  The  increase  in  non- 
availability or  the  loss  in  motivity,  is  thus  represented 
by  the  a,rea,fffd'd,  and  is  equal  to  the  increase  in  entropy 
multiplied  by  the  lowest  of  available  temperatures,  or 


X5H- 


This  increase  in  non-availability  is  thus  seen  to  be  equal 
to  the  heat  unavoidably  lost  under  the  original  coneli- 

m 

tions,  Tc-(h,  multiplied  by  a  factor  7/r^-l,   which   in- 

•L  8 

creases  in  value  as  Ta  decreases,  and  only  disappears 
when  T8  =  TF,  i.e.,  when  the  heat  is  used  at  the  tem- 
perature of  the  fire. 

Loss  of  Availability  Due  to  the  Liquid  Line. — Due  to 
the  fact  that  two  out  of  the  four  operations  constituting 


378        THE  TEMPERATURE-ENTROPY  DIAGRAM. 

the  Rankinc  cycle  occur  in  the  boiler  a  further  loss  must 
be  charged  against  it.  The  deviation  of  the  Rankine 
cycle  from  the  Carnot  cycle  has  already  been  discussed 
upon  pp.  233-234,  and  was  found  to  be  due  to  the  slope 
of  the  liquid  line  &s  determined  by  the  magnitude  of 
the  specific  heat  of  the  liquid  used. 

The  full  motivity  of  the  heat  as  determined  by  the 
boiler  pressure  is  not  possible  of  attainment,  as  part  of 
the  heat  absorbed  by  the  water  is  received  at  temper- 
atures less  than  that  corresponding  to  this  pressure. 
This  can  be  remedied  only  by  heating  the  feed  water 
to  boiler  temperatures  before  it  enters  the  boiler. 

The  deviation  between  the  Carnot  and  Rankinc  cycles 
for  any  given  case  can  be  readily  obtained  by  cornpu- 

T  —  T 
ting  the  motivity  —^—^  and   comparing   it  with   the 

* « 

corresponding  Rankine  efficiency  taken  from  the  table; 
on  p.  231. 

The  losses  due  to  the  boiler  may  therefore  be  sum- 
marized as 

(1)  Heat  lost  through  the  setting  and  up  the  stack. 

(2)  Loss  of  availability  due  to  reduced  temperature. 

(3)  Loss  of  availability  due  to  use  of  Rankine  cycle 
in  place  of  the  Carnot  cycle. 

The  Regenerative  Principle. — In  the  discussion  of  iso- 
diabatic  cycles  on  pp.  6-10  it  was  shown  that  the  effici- 
ency of  such  cycles  was  equal  to  that  of  the  Carnot  cycle, 
and  it  was  pointed  out  that  in  place  of  an  adiabatic 


ENTROPY  ANALYSIS  IN  THE  BOILER  ROOM.    379 

expansion  there  should  be  an  extraction  of  heat  by 
conduction  during  decrease  in  temperature  exactly  equal 
to  the  heat  required  during  the  rise-in-temperature  op- 
eration. (See  be  and  da,  Fig.  3.)  The  application  of 
this  principle  is  further  illustrated  on  pp.  158-160  in 
the  discussion  of  the  Stirling  and  Ericsson  cycles  for 
hot  air  engines. 

The  application  of  this  principle  to  the  cycle  of  water 
in  a  power  plant  necessitates  a  modification  of  the 


Fiu.  123. 

Rankine  cycle  whereby  the  adiabatic  expansion  line  is 
replaced  by  a  curve  whose  contour  is  a  duplicate  of  the 
water  line.  That  means  that  as  the-  steam  expands  an 
amount  of  heat  must  be  transferred  from  it  to  the  feed 
water  just  equal  in  amount  to  that  required  to  heat  the 
water  at  eacli  temperature.  Thus  in  Fijr.  123  the  isen- 
tropic  curve  be'  is  replaced  by  the  curve  be  posse>srng 
the  same  slope  as  the  water  line  ad. 

If  it  were  possible  to  extract  heat  from  the  steam  in 


380       THE   TEMPERATURE-ENTROPY  DIAGRAM. 


this  fashion  during  its  expansion  it  would  then  be  pos- 
sible to  realize  theoretically  at  least  the  maximum  avail- 
ability of  the  heat  in  the  boiler. 

A  complete  utilization  of  this  principle  is  not  feasible 
with  steam  engines  as  constructed,  but  in  the  case  of 
compound  engines  it  is  possible  to  approximate  it  by 
taking  sufficient  steam  from  each  receiver  to  warm  the 


FIG.  124. 

feed  water  up  to  the  temperature  of  the  receiver  steam. 
Thus  for  example  in  a  triple  expansion  engine  operating 
with  cold  feed  water,  Fig.  124,  the  heat  A  can  be  taken 
from  the  first  receiver  and  used  to  warm  the  feed  water 
from  b  to  a,  and  the  heat  B  can  be  taken  from  the 
second  receiver  and  serve  to  warm  the  feed  water  from 
G  (the  temperature  of  exhaust)  to  b,  and  finally  the 
heat  C  can  be  taken  from  the  exhaust  steam  and  used 
to  warm  the  feed  water  from  d  (the  temperature  in  the 
supply  pipe)  to  c. 


ENTROPY  ANALYSIS  IN  THE   BOILER  ROOM.    381 

Although  this  regenerative  principle  as  thus  applied 
may  seem  startling,  the  first  time  it  is  considered  a 
moment's  thought  will  show  that  one  is  already 
familiar  with  part  of  it  under  different  names.  Thus 
the  quantity  of  heat  C  is  saved  in  all  economical  plants 
with  a  primary  heater,  while  part  of  B  at  least  is  saved 
by  utilizing  the  exhaust  of  the  auxilliary  engines  which 
ordinarily  discharge  at  atmospheric  pressure.  And 
finally  the  equivalent  of  the  rest  of  B  and  part  of  A  is 
obtained  by  saving  some  of  the  heat  of  the  exhaust 
gases  with  an  economizer. 

The  use  of  the  economizer  is  not  truly  a  part  of  this 
regenerative  principle,  but  is  the  utilization  of  heat 
otherwise  wasted,  and  this  opens  up  the  interesting 
question  as  to  whether  it  is  better  to  use  small  heating 
surface  in  the  main  boiler  combined  with  a  good  econ- 
omizer and  no  regenerative  action  from  the  interme- 
diate cylinders,  or  to  make  the  heating  surface  of  the 
boiler  so  extensive  that  nothing  remains  for  the  econo- 
mizer to  do,  combined  with  regenerative  action  at  the 
various  intermediate  receivers.* 

The  Primary  Feed-water  Heater. — Ideally  it  would 
be  perfectly  feasible  to  return  the  condensate  or  an 
equal  amount  of  new  feed  water  to  the  boiler  at  the 
temperature  of  the  exhaust  steam;  practically  the  tem- 
perature is  -always  considerably  less  than  this  because 

*  Engineering,  1895,  pp.63,  97,  191.  Feed-Water  Heaters,  by 
Professor  A.  C.  Elliott. 


382        THE  TEMPERATURE-ENTROPY  DIAGRAM. 


of  heat  losses  due  to  radiation  from  the  feed  pipe  between 
the  heater  and  the  boiler,  or  between  the  primary  heater 
and  any  secondary  device  which  may  be  employed. 
The  use  of  a  primary  heater  is  based  upon  the  impos- 
sibility or  rather  the  practical  unfeasibility  of  reducing 
the  temperature  of  the  exhaust  steam  to  that  of  the 
cooling  water.  That  is,  the  small  increase  in  power  is 
offset  by  greater  increased  loss  in  the  condenser  plant. 


FIG.  125. 

In  the  case  of  engines  using  vapors  other  than  steam 
where  the  pressure  at  exhaust  is  still  above  atmospheric, 
the  gain  in  power  from  the  lowering  of  the  back  pres- 
sure line  justifies  the  maintenance  of  a  lower  temper- 
ature in  the  exhaust  condenser,  so  that  the  question 
of  a  primary  feed-fluid  heater  is  eliminated  both  from 
the  thermal  as  well  as  the  practical  point  of  view. 
(See  temperatures  quoted  on  pp.  255-257  for  the  S02 
and  ^Eihylamine  engines. 


ENTROPY  ANALYSIS  IN  THE  BOILER  ROOM.    383 

The  saving  to  be  credited  to  the  primary  heater  is 
the  decrease  in  the  amount  of  heat  absorbed  by  the 
water  in  the  boiler  occasioned  by  its  use.  Thus  if  the 
heater  raises  the  temperature  from  if  to  h  (Fig.  125) 
the  decrease  in  the  amount  of  heat  absorbed  in  the  boiler 
per  pound  of  steam  is 


while  the  total  heat  absorbed  per  pound  without  the 
heater  is 


and  therefore  the  efficiency  of  the  heater  is  equal  to 


The  denominator  of  this  expression,  assuming  the  steam 
to  be  dry  and  the  cold  feed  water  to  be  at  an  average 
temperature  of  70°  F.,  fluctuates  in  value  from  1130 
B.T.U.  per  Ib.  at  40  Ibs.  gage  to  1165  B.T.U.  per  Ib.  at 
330  Ibs.  gage,  or  undergoes  a  variation  of  less  than  four 
per  cent  for  a  temperature  range  of  140°  F.  Between 
100  Ibs.  gage  and  150  Ibs.  gage  the  variation  is  from 
1147  to  1156  B.T.U.  per  Ib.,  or  less  than  one  per  cent. 
For  rough  calculations  the  average  value  of  the  de- 
nominator may  be  assumed  as  1160  B.T.U.  per  Ib. 

The  value  of  the  numerator  shows  the  effectiveness 
of  the  feed  water  heater,  and  it  is  evident  that  the 
efficiency  is  directly  proportional  to  the  increase  in  the 


384        THE  TEMPERATURE-ENTROPY  DIAGRAM. 
temperature  of  the  feed  water  and  equals  approximately 
*  J;;  so  that  on  an  average  11.0°  F.  increase  in  the 

J.  J.OU 

temperature  of  the  feed  water,  coriesponds  to  one  per 
cent  saving  in  coal,  and  an  increase  of  100°  F.  corre- 
sponds to  a  saving  of  about  nine  per  cent  in  coal. 

This  thermal  saving  is  but  one  of  many  economies 
instituted  by  a  feed-water  heater.  The  heating  of  the 
feed  water  serves  to  precipitate  many  of  the  impurities 
in  the  boiler  feed  in  the  heaters,  whence  they  may  be 
easily  removed,  instead  of  in  the  less  easily  accessible 
boiler.  This  results  in  less  impairment  of  the  conduc- 
tivity of  the  heating  surface  and  effects  a  saving  in  fuel 
varying,  according  to  the  manufacturers,  anywhere 
from  fifteen  per  cent  to  fifty  per  cent  with  hard  feed 
waters,  while  the  average  saving  with  soft  waters  is 
about  ten  per  cent. 

A  further  saving  is  obtained  from  the  decreased  chill- 
ing of  the  boiler  from  the  introduction  of  cold  water, 
thus  diminishing  the  internal  stresses  due  to  temper- 
ature changes,  reducing  the  repair  bills  upon  the  boiler, 
and  prolonging  its  life. 

The  Secondary  Feed-water  Heater. — The  primary 
heater  operating  with  exhaust  steam  at  one,  two  or 
three  pounds  pressure  can  raise  the  temperature  of  the 
feed  water  only  to  102°,  126°,  or  142°  F.  With  con- 
densing engines  there  is  a  steam  consumption  by  the 
auxiliaries  of  about  ten  per  cent  that  of  the  main 


ENTROPY  ANALYSIS  IX    THE  BOILER  ROOM.     385 

engine  which  is  usually  exhausted  to  the  atmosphere 
at  212°  F.,  or  slightly  in  excess  of  this  temperature. 
It  is  customary  to  utilize  part  of  the  waste  heat  in 
this  exhaust  to  raise  the  temperature  of  the  feed  coming 
from  the  primary  heater  to  as  near  212°  F.  as  possible. 

Thus  in  the  case  of  three  pounds  back-  pressure  it  is 
possible  to  heat  the  feed  water  from  142°  F.  to  212°  F. 
or  through  70°  F.  On  the  assumption  of  ten  per  cent 
consumption  by  the  auxiliaries,  11  pounds  of  feed  water 
must  be  heated  for  each  pound  used  by  the  auxiliaries, 
i.e.,  770  B.T.U.  would  need  to  be  supplied  by  the  secon- 
dary heater.  Each  pound  of  steam  at  212°  F.  possesses 
a  heat  of  condensation  equal  to  970  B.T.U.,.  so  that,  assum- 
ing the  exhaust  from  the  auxiliaries  to  be  fifteen  per 
cent  condensed,  this  leaves  970  X.  85  -823  B.T.U.  avail- 
able for  heating,  which  is  more  than  ample.  In  case 
the  main  engine  exhausts  at  lower  pressure  a  greater 
heat  demand  is  made  upon  the  secondary  heater,  but 
this  adjusts  itself  automatically  to  a  considerable  extent 
by  the  increased  steam  consumption  of  the  auxiliaries. 

The  total  saving  due  to  primary  and  secondary 
heaters  expressed  in  per  cent  of  the  heat  required  by 
the  steam  without  such  heaters  is  given  by 


which,  upon  the  assumption  of  70°  F.  as  the  average 
temperature  of  the  supply  water,  reduces  to 


386       THE   TEMPERATURE-ENTROPY  DIAGRAM. 
142 


.122, 


1160 
or  12.2  per  cent  at  ordinary  boiler  pressures. 

Saving  Effected  by  an  Economizer. — Advantage  may 
be  taken  of  the  unavoidable  loss  of  heat  up  the  stack 
to  offset,  at  least  partially,  the  loss  caused  by  cold  feed 
water.  Although  the  hot  gases  cannot  be  cooled  below 
the  temperature  of  the  boiler  and  must  in  fact  be  con- 
siderably hotter  than  the  steam  to  insure  rapid  flow  of 
heat,  they  are  still  hot  enough  to  impart  considerable 
heat  to  the  colder  feed  water.  This  is  accomplished  by 
passing  the  feed  water  on  its  way  to  the  boiler  through 
suitable  piping  installed  in  the  path  of  the  hot  gases. 

Such  an  economizer  will  not  deliver  the  water  to  the 
boiler  at  boiler  temperature,  but  will  succeed  in  raising 
it  far  above  the  temperatures  obtained  in  primary  and 
secondary  heaters.  For  example,  tests  on  Green  econ- 
omizers show  an  increase  in  the  feed  temperature  of 
about  120°  F.,  whether  the  water  enters  the  economizer 
at  40°  F.  or  200°  F.  making  but  little  difference.  The 
gases  leaving  the  economizer  possessed  temperatures 
varying  from  254°  F.  to  293°  F. 

Any  such  increase  in  temperature  above  that  of  the 
secondary  heater  represents  a  direct  saving  of  heat 
which  would  otherwise  be  wasted.  It  might  seem  pos- 
sible at  first  sight  to  bring  the  feed  water  up  to  the 
boiler  temperature,  but  this  is  impracticable  as  a  cer- 
tain temperature  of  the  exhaust  gases  is  required  to 


EXTROPY   ANALYSIS  IN   THE  BOILER  ROOM.    387 

produce  the  necessary  draft  in  the  chimney.  If  cooled 
beyond  this  point  it  becomes  necessary  to  install  a 
forced  draft  and  then  its  steam  cost  must  be  charged 
against  the  saving  produced  by  the  economizer. 

In  many  installations  economizers  are  installed  in 
place  of  secondary  and  sometimes  even  in  place  of 
primary  heaters,  and  in  such  case  they  obtain  credit 
for  savings  due  to  other  devices,  but  apparently  they 
produce  about  the  same  increase  in  temperature  re- 
gardless of  the  initial  temperature  of  the  water. 


TABLE  OF  PROPERTIES. 


PROPERTIES  OF  SATURATED  STEAM  FROM  400°  F.  TO 
THE  CRITICAL  TEMPERATURE.* 


Increase 

in 

Heat  of 

Temper- 
ature, 
Degrees 

Absolute 
Pressure 
in  Lbs. 

Specific 
Volume 
Due  to 

the 
Liquid 

Latent 
Heat  in 
B.T.U. 

Total 
Heat. 

Entropy 
Water 

Entropy 

xr  °f 

Vapor- 

Fahren- 
heit. 

per 
SqP  In. 

Vapori- 
zation in 
Cu.  Ft. 

per  Lb. 

per  Lb. 

above 
32°  F. 

ization. 

t 

P 

a  —  a 

1 

r 

fl+r 

4»-o 

*-* 

420 

308.7 

1.495 

390.1 

818.9 

1209.0    .5912 

.9254 

430 

343.6 

1.347 

405.6 

805.4 

1211  0 

.6031 

.9055 

440 

381.5 

1.215 

416.1 

796.2 

1212.3 

.6118 

.8853 

450 

422.4 

1.099 

426.7 

786.5 

1213.2 

.6266 

.8648 

460 

466.2 

.9953 

437.3 

777.4 

1214.7 

.6381 

.8455 

470 

513.5 

.9024 

447.9 

765.2 

1213.1 

.6495 

.8233 

480 

564.4 

.8194 

458.5 

753.5 

1212.0 

.6610 

.8021 

490 

619.5 

.7436 

469.2 

741.3 

1210.5 

.6723 

.7808 

500 

678.5 

.6752 

479.9 

728.2 

1208.1 

.6835 

.7590 

510 

741.7 

.6134 

490.6 

714.4 

1205.0 

.6946 

.7369 

520 

808.5 

.5565 

501.4 

699.8 

1201.2 

.7058 

.7145 

530 

889.1 

.5050 

512.1 

684.4 

1196.5 

.7166 

.6917 

540 

956.1 

.4585 

523.0 

668.1 

1191.1 

.7275 

.6685 

550 

1036. 

.4174 

533.8 

650.8 

1184.6 

.7383 

.6448 

560 

1122. 

.3798 

544.7 

632.4 

1177.1 

.7490 

.6204 

570 

1212. 

.3457 

555.6 

612.9 

116S.  5 

.7597 

.5955 

580 

1308. 

.3131 

566.5 

592.2 

1158.7 

.7702 

.5698 

590 

1409. 

.2828 

577.5 

570.0 

1147.5 

.7807 

.5432 

600 

1516. 

.2547 

588.5 

546.3 

1134.8 

.7912 

.5157 

610 

1628. 

.2283 

599.6 

520.8 

1120.4 

.8015 

.4871 

620 

1745. 

.2037 

610.6 

493.2 

1103.8 

.8118 

.4569 

630 

1867. 

.1809 

621.7 

463.2 

1084.9 

.8221 

.4252 

640 

2000. 

.1610 

632.9 

430.3 

1063.2 

.8322 

.3914 

650 

2137. 

.1416 

641.0 

396.7 

1037.7 

.8422 

.3549 

660 

2281. 

.1217 

655.2 

352.2 

1007.4 

.8527 

.3146 

670 

2431  . 

.  1031 

666.5 

304.1 

970.6 

.8634 

.2693 

680 

2586. 

.0800 

677.8 

245.2 

923.0 

.8723 

.2152 

690 

2748. 

.0464 

689.1 

164.3 

853.4 

.8825 

.1429 

698 

2882. 

0 



0 



0 

*  See  Engineering  (London),  Jan.  4,  1907. 


HYPERBOLIC  OR  NAPERIAN  LOGARITHMS. 

THE  log  of  a  number  is  the  exponent  of  the  power 
to  which  it  Is  necessary  to  raise  a  fixed  number,  called 
the  base,  to  produce  the  given  number.  In  the  com- 
mon logs  the  base  is  10  and  in  the  Naperian  logs  the 
base,  e,  is  2.71828  .  .  .  The  mathematical  relations 
between  these  two  systems  and  any  third  are  given 
by  the  equations, 

lQlog]0« 


where  a  is  any  number  and  b  is  the  third  base. 
Whence 

loge  a  =  loge  10-logio  a  =  2.3026  logic  a 
and 

logio  a  =  logio  e-loge  a  =  0.4343  loge  a 
and 

ea    or   = 


In  general,  the  log  of  a  number  in  any  system  equals 
either  the  reciprocal  of  the  Naperian  log  of  the  base 
of  that  system  times  the  Naperian  log  of  the  number, 
or  equals  the  reciprocal  of  the  common  log  of  the 
base  of  that  system  times  the  common  log  of  the 
number. 

In  any  system  the  base  of  which  is  greater  than 
1,  the  logs  of  all  numbers  greater  than  1  are  positive 

and  the  logs  of  all  numbers  less  than  1  are  negative. 

389 


390       THE  TEMPERATURE-ENTROPY  DIAGRAM. 


NAPERIAN    LOGARITHMS. 

e=2.7182818  log  e= 0.4342945=  M. 


0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

0.08618 

1.0 

0.0000 

0.00995 

0.01980 

0.02956 

0.03922 

0.04879 

0.05827 

0.06766 

0.07696 

'.2 
1.3 

0.09531 
0.1823 
0.2624 

0.1044 
0.1906 
0.2700 

0.1133 
0.1988 
0.2776 

0.1222 
0.2070 
0.2852 

0.1310 
0.2151 
0.2927 

0.1398 
0.2231 
0.3001 

0.1484 
0.2311 
0.3075 

0.1570 
0.2390 
0.3148 

0.1655 
0.2469 
0.3221 

0.1739 
0.2546 
0.3293 

1.4 
1.5 
1.6 

0.3365 
0.4055 
0.4700 

0.3436 
0.4121 
0.4762 

0.3507 
0.4187 
0.4824 

0.3577 
0.4243 
0.4886 

0.3646 
0.4318 
0.4947 

0.3716 
0.4382 
0.5008 

0.3784 
0.4447 
0.5068 

0.3853 
0.4511 
0.5128 

0.3920 
0.4574 
0.5188 

0.3988 
0.4637 
0.5247 

17 

1.8 
1.9 

0.5306 

0.5878 
0.6418 

0.5365 

1  -,((•;•; 

L6471 

0.5423 
0.5988 
0.6523 

0.5481 
0.6043 
0.6575 

0.5539 
0.6098 
0.6627 

0.5596 
0.6152 
0.6678 

0.5653 
0.6206 
0.6729 

0.5710 

0.6259 
0.6780 

0.5766 
0.6313 
0.6831 

0.5822 
0.6366 
0.6881 

2.0 

0.6931 

0.6981 

0.7031 

0.7080 

0.7129 

0.7178 

0.7227 

0.7275 

0.7324 

0.7372 

2.1 

2.2 
2.3 

0.7419 

0.7884 
0.8329 

0.7467 
0.7930 
0.8372 

0.7514 
0.7975 
0.8416 

0.7561 
0.8020 
0.8459 

0.7608 
0.8065 
0.8502 

0.7655 
0.8109 
0.8544 

0.7701 
0.8154 
0.8587 

0.7747 
0.8198 
0.8629 

0.7793 
0.8242 
0.8671 

0.7839 
0.8286 
0.8713 

2.4 

1  5 
2.6 

0.8755 
0.9163 
0.9555 

0.8796 
0.9203 
0.9594 

0.8838 
0.9243 
0.9632 

0.8879 
0.9282 
0.9670 

0.8920 
0.9322 
0.9708 

0.8961 
0.9361 
0.9746 

0.9002 
0.9400 
0.9783 

0.9042 
0.9439 
0.9821 

0.9083 
0.9478 
0.9858 

0.9123 
0.9517 
0.9895 

2'S 
2  «) 

0.9933 
1.0296 
1.0647 

0.9969 
1.0332 
1.0682 

1.0006 
1.0367 
1.0716 

1.0043 
1.0403 
1.0750 

1.0080 
1.0438 
1.0784 

1.0116 
1.0473 
1.0818 

1.0152 
1.0508 
1.0852 

1.0188 
1.0543 
1.0886 

.0225 
.0578 
1.0919 

1.0260 
1.0613 
1.0953 

3.0 

1.0986 

1.1019 

1.1053 

1.1086 

1.1119 

1.1151 

1.1184 

1.1217 

1.1249 

1.1282 

LI 

3^3 

1.1314 
1.1632 
1.1939 

1.1346 
1.1663 
1.1969 

1.1378 
1.1694 
1.2000 

.1410 
.1725 
.2030 

1.1442 
1.1756 
1.2060 

1.1474 
1.1787 
1.2090 

1.1506 
1.1817 
1.2119 

1.1537 
1.1848 
1.2149 

.1569 
1.1878 
.2179 

1.1600 
1.1909 
1.2208 

!V5 
.56 

1.2238 
1.2528 
1.2809 

1.2267 
1.2556 
1.2837 

1.2296 
1.2585 
1.2865 

.2326 
.2613 
1.2892 

1.2355 
1.2641 
1.2920 

1.2384 
1.2669 
1.2947 

1.2413 

1.2698 
1.2975 

1.2442 
1.2726 
1.3002 

.2470 

.L'754 
1.3029 

1.2499 

1.2782 
1.3056 

3.7 
3  8 
3.'> 

1.3083 
1.3350 
1.3610 

1.3110 
1.3376 
1.3635 

1.3137 
1.3403 
1.3661 

1.3164 
1.3429 
1.3686 

1.3191 
1.3455 
1.3712 

1.3218 
1.3481 
1.3737 

1.3244 
1.3507 
1.3762 

1.3271 
1.3533 
1.3788 

1.3297 
.3558 
1.3813 

1.3324 
1  .3584 
1.3838 

1.0 

1.3863 

1.3888 

1.3913 

1.3938 

1.3962 

1.3987 

1.4012 

1.4036 

1.4061 

1.4085 

1.1 
4.2 
4.3 

1.4110 
1.4351 
1.4586 

1.4134 
1.4375 
1.4609 

1.4159 
1.4398 
1.4633 

1.4183 
1.4422 
1.4656 

1.4207 
1.4446 
1.4679 

.4231 
.4469 
.4702 

1.4255 
1.4493 
1.4725 

1.4279 
1.4516 
1.4748 

1.4303 
1.4540 
1.4770 

1.4327 
1.4563 
1.4793 

4.4 
1  5 
4.6 

1.4816 
1.5041 
1.5261 

1.4839 
1.5063 
1.5282 

1.4861 
1.5085 
1.5304 

1.4884 
1.5107 
1.5326 

1.4907 
1.5129 
1.5347 

.4929 
.5151 
.5369 

.4951 
.5173 
.5390 

1.4974 
1.5195 
1.5412 

1.4996 
1.5217 
1.5433 

1.5019 
1.5239 
1.5454 

4.7 

4.8 
4.< 

1.5476 
1.5686 
1.5892 

1.5497 
1.5707 
1.5913 

1.5518 
1.5728 
1.5933 

1.5539 
1.5748 
1.5953 

1.5560 
1.5769. 
1.5974 

.5581 
.5790 
1.5994 

.5602 
.5810 
1.6014 

1.5623 
1.5831 
1.6034 

1.5644 
1.5851 
1.6054 

1.5665 
1.5872 
1.6074 

5.( 

1.6094 

1.6114 

L6134 

1.6154 

1.6174 

1.6194 

1.6214 

1.6233 

1.6253 

1.6273 

5. 
5.2 
5.. 

1.6292 
1.6487 
1.6677 

1.6312 
1.6506 
1.6696 

1.6332 
1.6525 
1.6715 

1.6351 

1.6544 
1.6734 

1.6371 
1.6563 
1.6752 

1.6390 
1.6582 
1.6771 

1.6409 
1.6601 
1.6790 

1.6429 
1.6620 
1.6808 

1.6448 
1.0639 
1.6827 

1.6467 

1  .66.58 
1.6845 

5, 
5.5 
56 

1.6864 
1.7047 
1.7228 

1.6882 

l.Tonr, 
1.7246 

1.6901 
1.7084 
1.7263 

1.6919 
1.7102 
1.7281 

1.6938 
1.7120 
1.7299 

1.6956 
1.7138 
1.7317 

1.6974 
1.7156 
1.7334 

1.6993 
1.7174 
1.7354 

1.7011 
1.7192 
1.7370 

1.7029 
1.7210 
1.7387 

HYPERBOLIC  OR  NAPER1AN  LOGARITHMS.      391 
XAPERIAN    LOGARITHMS— Continued. 


5.7 
5.8 
5.9 

0 

1 

2 

3 

4 

5 

6 

7 

8 

9 

1.7405 
1.7579 
1.7750 

1.7422 
1.7596 
1.7766 

1.7440 
1.7613 
1.7783 

1.7457 
1.7630 
1.7800 

1.7475 
1.7647 
1.7817 

1.7492 
1.7664 
1.7834 

1.7509 
1.7681 
1.7851 

1.7527 
1.7699 
1.7867 

1.7544 
1.7716 
1.7884 

1.7561 
1.7733 
1.7901 

6.0 

1.7918 

1.7934 

1.7951 

1.7967 

1.7984 

1.8001 

1.8017 

1.8034 

1.8050 

1.8066 

6.1 
6.2 
6.3 

1.8083 
1.8245 
1.8405 

1.8099 
1.8262 
1.8421 

1.8116 
1.8278 
1.8437 

1.8132 
1.8294 
1.8453 

1.8148 
1.8310 
1.8469 

1.8165 
1.8326 
1.8485 

1.8181 
1.8342 
1.8500 

1.8197 
1.8358 
1.8513 

1.8213 
1.8374 
1.8532 

1.8229 
1.8390 
1.8547 

6.4 
6.5 
6.6 

1.8563 
1.8718 
1.8871 

1.8579 
1.8733 
1.8886 

1.8594 
1.8749 
1.8901 

1.8610 
1.8764 
1.8916 

1.8625 
1.8779 
1.8931 

1.8641 
1.8795 
1.8946 

1.8656 
1.8810 
1.8961 

1.8672 
1.8825 
1.8976 

1.8687 
1.8840 
1.8991 

1.8703 
1.8856 
1.9006 

6.7 

6.8 
6.9 

1.9021 
1.9169 
1.9315 

1.9036 
1.9184 
1.9330 

1.9051 
1.9199 
1.9344 

1.9066 
1.9213 
1.9359 

1.9081 
1.9228 
1.9373 

1.9095 
1.9242 
1.9387 

1.9110 
1.9257 
1.9402 

1.9125 
1.9272 
1.9416 

1.9140 

1.9286 
1.9430 

1.9155 
1.9301 
1.9445 

7.0 

1.9459 

1.9473 

1.9488 

1.9502 

1.9516 

1.9530 

1.9544 

1.9559 

1.9573 

1.9587 

7.1 

7.2 
7.3 

1.9601 
1.9741 
1.9879 

1.9615 
1.9755 
1.9892 

1.9629 
1.9769 
1.9906 

1.9643 
1.9782 
1.9920 

1.9657 
1.9796 
1.9933 

1.9671 
1.9810 
1.9947 

1.9685 
1.9824 
1.9961 

1.9699 
1.9838 
1.9974 

1.9713 
1.9851 
1.9988 

1.9727 

1.9865 
2.0001 

7.4 

7.5 
7.6 

2.0015 
2.0149 
2.0281 

2.0028 
2.0162 
2.0295 

2.0042 
2.0176 
2.0308 

2.0055 
2.0189 
2.0321 

2.0069 
2.0202 
2.0334 

2.0082 
2.0215 
2.0347 

2.0096 
2.0229 
2.0360 

2.0109 
2.0242 
2.0373 

2.0122 
2.0255 
2.0386 

2.0136 
2.0268 
2.0399 

7.7 
7.8 
7.9 

2.0412 

_Mi:>41 
2.0668 

2.0425 
2.0554 
2.0681 

2.0438 
2.0567 
2.0694 

2.0451 
2.0580 
2.0707 

2.0464 
2.0592 
2.0719 

2.0477 
2.0605 
2.0732 

2.0490 
2.0618 
2.0744 

2.0503 
20631 
2.0757 

2.0516 
2.0643 
2.0769 

2.0528 
2.0656 
2.0782 

8.0 

2.0794 

2.0807 

2.0819 

2.0832 

2.0844 

2.0857 

2.0869 

2.0881 

2.0894 

2.0906 

8.1 
82 
8.3 

2.0919 
2.1041 
2.1163 

2.0931 
2.1054 
2.1175 

2.0943 
2.10M 

2.1187 

2.0956 
2.1078 
2.1199 

2.0968 
2.1090 
2.1211 

2.0980 
2.1102 
2.1223 

2.0992 
2.1114 
2.1235 

2.1005 
2.1126 
2.1247 

2.1017 
2.1138 
2.1258 

2.1029 
2.1150 
2.1270 

8.4 
8.5 
8.6 

2.1282 
2.1401 
2.1518 

2.1294 
2.1412 
2.1529 

2.1306 
2.1424 
2.1541 

2.1318 

•2  n:j,r, 
LM  ,-,.->:.' 

2.1330 
2.1448 
2.1564 

2.1342 
2.1459 
2.1576 

2.1353 
2.1471 
2.1587 

2.1365 
2.1483 
2.1599 

2.1377 
2.1494 
2.1610 

2.1389 
2.1506 
2.1622 

8.7 

KM 

8.9 

2.1633 
2.1748 
2.1861 

2.1645 
2.1759 
2.1872 

2.1656 
2.1770 
2.1883 

2.1668 
2.1782 
2.1894 

2.1679 

•2  17M 
2.1905 

2.1691 
2.1804 
2.1917 

2.1702 
2.1815 
2.1928 

2.1713 
2  1827 
2.1939 

2  1725 
2.1838 
2.1950 

2.1736 
2.1849 
2.1961 

9.0 

2.1972 

2.1983 

2.1994 

2.2006 

2.2017 

2.2028 

2.2039 

2.2050 

2.2061 

2.2072 

9.1 
92 
9.3 

2.2083 

2.2192 
2.2300 

2.2094 
2.2203 
2.2311 

2.2105 
2.2214 
2.2322 

2.2116 
2.2225 
2.2332 

2.2127 
2.2235 
2.2343 

2.2138 
2.2246 
2.2354 

2.2148 
2.2257 
2.2364 

2.2159 
2.2268 
2.2375 

2.2170 
2.2279 
2.2386 

2.2181 

2.2289 
2.2396 

9.4 
9.5 
9.6 

2.2407 
2.2ol3 
2.2618 

2.2418 
2.2523 
2.2628 

2.2428 
2.2534 
2.2638 

2.2439 
2.2544 
2.2649 

2.2450 
2.2555 
2.2659 

2.2460 
2.2565 
2.2670 

2.2471 
2.2576 
2.2680 

2.2481 
2.25M 

2.2690 

2.2492 
2.2597 
2.2701 

2.2502 
2.2607 
2.2711 

9.7 
9.8 
9.9 

2.2721 

2.2824 
2.2925 

2.2732 
2.2834 
2.2935 

2.2742 
2.2844 
2.2946 

2.2752 
2.2854 
2.2956 

2.2762 

2.2865 
2.2966 

2.2773 
2.2875 
2.2976 

2.2783 

LV2.S.S.-, 
2.2986 

2.2793 

2.2895 
2.2996 

2.2803 
2.2905 
2.3006 

2.2814 
•j  NIB 

2.3016 

10.0 

2.3026 

392       THE    TEMPERATURE-ENTROPY  DIAGRAM. 
LOGARITHMS. 


Nat.  Nos. 

0 

I 

2 

3 

4 

5 

6 

7 

8 

9 

Proportional  Parts. 

1  23 

456 

789 

10 
11 
12 

0000 
0414 
0792 

0043 
0453 

0828 

0086 
0492 
0864 

0128 

0.531 
0899 

0170 
0569 
0934 

0212 
0607 
0969 

0253 

064.5 
L004 

0294 
0682 
103s 

0334 

071!) 

1072 

0374 

07.5.5 
1106 

4  8  12 
4  8  11 
3  7  10 

17  21  25 
15  19  23 
14  17  21 

29  33  37 
26  30  34 
24  28  31 

13 

1139,1173 

1206  1239 

1271 

1303 

1  :;:,.-, 

1367 

1:599 

3  6  10 

13  16  19 

23  26  29 

14 

1461 

1492 

1523,1553 

1584 

1614 

1644 

1673 

1703 

1732 

369 

12  15  18 

21  24  27 

15 

1761 

1790 

18181847 

187.5 

1903 

1931 

1959 

1987 

2014 

368 

11  14  17 

20  22  25 

16 

2041 

2068 

2095:2122 

2148 

217.5 

2201 

2227 

22.53 

22",  9 

3  5    8 

11  13  16 

18  21  24 

17 

230412330 

23552380 

240,5 

2430 

245512480 

2504 

2,529 

257 

10  12  15 

17  20  22 

18 

2553 

2,577 

2601  2625 

264s 

2:172 

2695 

271s 

2742 

276,5 

257 

9  12  14 

16  19  21 

19 

2788 

2810 

2833 

2856 

2878 

2900 

292:i 

294,5 

2967 

2989 

247 

9  11  13 

16  18  20 

20 

3010 

3032 

3054 

3075 

509f 

3118 

3139 

3160 

3181 

3201 

246 

8  11  13 

15  17  19 

21 

3222 

32  n 

326213284 

5304 

3324 

334,5 

:-',365 

;::s,5 

3404 

246 

8  10  12 

14  16  18 

22 

1424 

3411 

34643483 

5,502 

3522 

3.541 

:!,560 

',.579 

3.59s 

246 

8  10  12 

14  15  17 

23 

3617 

3636 

3655!3674 

I'll): 

3711 

3729 

3717 

3700 

3784 

246 

7    9  11 

13  15  17 

24 

3802 

3820 

3838 

3856 

3874 

3892 

3909 

3927 

394,5 

3962 

245 

7    9  11 

12  14  16 

25 

3979 

3997 

4014 

4031 

4048 

406,5 

4082 

4099 

4116 

4133 

235 

7    9  10 

12  14  15 

26 

11,50 

4166 

4183 

4200 

12  ic 

4232 

1219 

1205 

1281 

4298 

235 

7    8  10 

11  13  15 

27 

1314 

i:i:;o 

f.'.Kl 

4362 

l:;7s 

4393 

4409 

1425 

4410 

4  t,5(l 

2  3    5 

689 

11  13  14 

28 

1172 

1187 

4502 

4518 

!.i:;:; 

1.54s 

4564 

4579 

1591 

4609 

235 

689 

11  12  14 

29 

4624 

1639 

4654 

4669 

1683 

4698 

4713 

4728 

4742 

4757 

1  3    4 

679 

10  12  13 

30 

4771 

4786 

4800 

4814 

4829 

4843 

4857 

4871 

4886 

4900 

3    4 

6    7    9 

10  11  13 

31 

4914 

4928 

4942 

4955 

4969 

19S.:; 

4997 

,5011 

502.1 

.5038 

3    4 

678 

10  11  12 

32 

5051 

S005 

5079 

5092 

5105 

.5119 

5132 

51  1.5 

51.59 

5172 

3 

578 

9  11  12 

33 

518,5 

519s 

5211 

5224 

5237 

52(1:; 

527(1 

5289 

,5302 

3 

568 

9  10  12 

34 

5315 

5328 

5340 

5353 

5366 

.5378 

5391 

5403 

541(1 

5428 

3 

568 

9  10  11 

35 

5441 

5453 

5465 

5478 

5490 

5.502 

5514 

5,527 

5.5:59 

5551 

2 

5    6    7 

9  10  11 

36 

6603 

5575 

5S87 

:,599 

5611 

5623 

563,5 

5647 

56.58 

5670 

2 

567 

8  10  11 

37 

5682 

5994 

5705 

5717 

5729 

57.10 

5752 

5763 

577.5 

.578(1 

2    3 

567 

8    9  10 

38 

5798 

5809 

5821 

5832 

5843 

58.55 

5866 

5877 

5S88 

5S99 

2    3 

567 

8    9  10 

39 

5911 

5922 

5933 

5944 

59,5.5 

5966 

5977 

59XS 

5999 

6010 

2    3 

457 

8    9  10 

40 

6021 

5031 

6042 

6053 

6064 

607,5 

608.5 

H096 

6107 

6117 

2    3 

5     6 

8    9  10 

41 

612x 

1138 

1149 

6160 

6170 

5180 

1191 

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789 

42 
43 
44 

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(ll.V, 

121:; 
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6444 

125:; 
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14.54 

6263 
r,.;r,5 
6464 

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137.5 
6474 

6284 

i:;s.-, 
1484 

1291 
639.5 

6493 

6304 
1405 

6314 
1415 

6,513 

!i:;25 
6425 
6522 

2    3 
2    3 
2    3 

5     6 
5     6 
5     6 

789 
789 
7    8    9 

45 
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0632 

6628 

1542 
66:57 

6,5.51 

6646 

6561 

iiti.56 

6571 

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167.5 

6.590 

K1S! 

6,599 
6693 

6609 
6702 

6618 

6712 

2    3 
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5    6 

789 
778 

47 

6721 

1730 

6739 

6719 

67.5S 

6767 

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678.5 

1791 

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2    3 

5     5 

678 

48 
49 

(IS  12 
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1911 

1830 

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6839 
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1937 

is:,  7 
194-1 

1800 

1955 

687.5 
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1972 

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6981 

2    3 
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4    5 

6    7    '8 
678 

50 
51 

6990 
7076 

6998 

70S1 

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7093 

7016 
7101 

7024 
7110 

7033 

7118 

7042 
7126 

70.50 
713.5 

70.59 
711:; 

7067 

7152 

2    3 
2    3 

3     4     5 
345 

678 
678 

52 

7160 

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7177 

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2    2 

345 

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54 

7324 

7332 

7340 

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7364 

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COMMON   LOGARITHMS. 


393 


LOGARITHMS— Continued. 


"6 
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Proportional  Parts. 

"ej 
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0 

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9 

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58 

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344     567 

59 

7709 

7716 

7723 

7731 

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1  12 

344     567 

60 

7782 

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7875 

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344     566 

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7924 

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7966 

7973 

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1  1  2 

334     566 

63 

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8000 

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1  1  2 

334556 

64 

8062 

8069 

8075 

8082 

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1  1  2 

334     556 

65 

8129 

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8149 

8156 

8162 

8169 

8176 

8182 

8189 

112 

334     556 

66 

8195 

8202 

8209 

8215 

8222 

8228 

8235 

8241 

8248 

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1  1  2 

334     556 

67 

8261 

8267 

8274 

8280 

8287 

8293 

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8319 

1  1  2 

334    556 

68 

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8370 

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1  1  2 

3341456 

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112 

234    456 

70 

8451 

8457 

8463 

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8488 

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1  1  2 

23  4  j  456 

71 

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University  of  California 

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405  Hilgard  Avenue,  Los  Angeles,  CA  90024-1388 

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